MATHEMATICS 
PHILOSOPHY 


CASSIUS  J.  KEYSER 


LIBRARY 

UNIVERSITY  OF  CALIFORNIA 

RIVERSIDE 


MATHEMATICAL  PHILOSOPHY 


BY    THE    SAME    AUTHOR 


Science  and  Religion:  The  Rational 
and  the  Supexrational. 

— The  Yale  University  Press. 

The  New  Infinite  and  the  Old 
Theology. 

— The  Yale  University  Press. 

The   Human  Worth   of  Rigorous 
Thinking. 
— The  Columbia  University  Press. 


MATHEMATICAL  PHILOSOPHY 

A  STUDY  OF  FATE  AND  FREEDOM 

Lectures  for  Educated  Laymen 

BY 

CASSIUS  J.  KEYSER,  Ph.D.,  LL.D. 

Adratn  Professor  of  Mathematics  in  Columbia  University 


NEW  YORK 
E.  P.  DUTTON  &  COMPANY 

68 1   FIFTH  AVENUE 


Copyright,  1922, 
By  E.  P.  DUTTON  &  COMPANY 


All  Rights  Rturtid 


First  Printing,    .     .     Feb.,  1922 
Second  Printing,     .     July,  1924 


FrinUd  in  th*  UniUd  Statts  of  America 


TO 


WITHDRAWN 


PREFACE 


For  more  than  two  score  years  I  have  meditated  upon 
the  nature  of  Mathematics,  upon  its  significance  in 
Thought,  and  upon  its  bearings  on  human  Life.  In  the 
following  course  of  lectures  I  have  endeavored  to  present, 
in  the  language  current  among  educated  men  and  women, 
some  of  the  maturer  fruits  of  that  study. 

Though  the  course  is  designed  primarily  for  students 
whose  major  interest  is  in  Philosophy,  I  venture  to  hope 
that  the  lectures  may  not  be  ungrateful  to  a  much  wider 
circle  of  readers  and  scholars: 

To  the  growing  class  of  such  professional  mathemati- 
cians as  are  not  without  interest  in  the  philosophical 
aspects  of  their  science. 

To  the  growing  class  of  such  teachers  of  mathematics 
as  endeavor  to  make  the  spirit  of  their  subject  dominate 
its  technique. 

To  the  growing  class  of  those  natural-science  students 
who  are  interested  in  the  logical  structure  and  the  dis- 
tinctive method  of  mathematics  regarded  not  only  as  a 
powerful  instrument  for  natural  science  but  also  and 
especially  as  the  prototype  which  every  branch  of  science 
approximates  in  proportion  as  its  basal  assumptions  and 
concepts  become  clearly  defined. 

To  the  innumerous  but  precious  tribe  of  those  literary 
critics  who  know  that  the  art  of  Criticism  owes  its  first 
allegiance  to  the  eternal  laws  of  thought. 

To  such  psychologists  as  are  interested  either  in  the 

vii 


viii  PREFACE 

psychology  of  mathematics   or  in   the   mathematics   of 
psychology. 

To  such  sociologists  as  desire  to  conceive  the  nature  of 
our  humankind  justly — in  accord  with  the  mathematical 
principle  of  "  logical  types  "  or  dimensionality. 

To  the  rapidly  increasing  class  of  engineers  who  are 
learning  to  conceive  engineering  worthily,  as  the  science 
and  art  of  directing  the  civilizing  energies  of  the  world 
to  the  advancement  of  the  welfare  of  all  mankind  including 
posterity. 

Finally,  to  all  readers  who  desire  to  acquire  a  fair 
understanding  of  such  genuinely  great  mathematical 
ideas  as  are  accessible  to  all  educated  laymen  and  to 
come  thus  into  touch  with  the  universal  spirit  of  the 
science  which  Plato  called  divine. 

In  closing  this  preface  I  desire  to  record  my  gratitude 
to  Mr.  John  Macrae,  vice-president  of  E.  P.  Dutton  & 
Company,  for  his  generous  encouragement  in  this  enter- 
prise. 

Cassius  Jackson  Keyser. 

Columbia  University, 

New  York,  January  II,  1922. 


CONTENTS 


LECTURE  I 

Introduction 


pages 


Intellectual  Freedom  and  Logical  Fate — Mathematical  Obli- 
gations of  Philosophy  and  Education — Common  Human- 
ity and  Individuality — Humanistic  and  Industrial  Edu- 
cation— Man  Not  an  Animal — Ethics  Not  a  Branch  of 
Zoology — Excellence  and  the  Muses — Logic  the  Muse  of 
Thought — The  Heroic  Tradition  in  Philosophy — Radiant 
Aspects  of  an  Overworld 1-38 

LECTURE  II 

Postulates 

Concrete  Definition  of  Postulate  System — The  Prototype  of 
Principles  or  Platforms — The  Ancient  "  Craft  of  Gem- 
etry" — The  Sword  of  the  Gadfly — Clarity  or  Silence — 
Municipal  Laws  and  the  Laws  of  Thought 39~48 

LECTURE  III 

Basic  Concepts 

Propositional  Function  and  Doctrinal  Function — Marriage  of 
Matter  and  Form — Its  Infinite  Fertility — Propositions 
and  Doctrines  the  Offspring — Verifiers  and  Falsifiers — 
Significance    and    Non-sense — A    Question    Asked    by 

Many  and  Answered  by  None 49~59 

ix 


x  CONTENTS 

LECTURE  IV 

Doctrinal  Interpretations 

pages 
A  Mother  of  Doctrines  Mistaken  for  Her  Eldest  Child — 
Infinitely  Many  Interpretations  of  One  Doctrinal 
Function — Ordinary  Geometry  but  One  of  Them — Other 
Interpretations  Geometric,  Algebraic  and  Mixed — 
Identity  of  Form  with  Diversity  of  Content — Distinction 
of  Logical  and  Psychological — Projective  Geometry  the 
Child  of  Architecture — A  Science  Born  of  an  Art — Infinite 
Points  and  the  Meeting  of  Parallels — Pole-to-Polar 
Transformations — Logical  Use  of  Pathological  Config- 
urations        60-72 

LECTURE  V 

Another  Geometric  Interpretation 

Brief  Introduction  to  the  Method  of  Descartes  and  Fermat — 
Inversion  Geometry  and  Inversion  Transformation — The 
Infinite  Region  of  Inversion  Space  a  Point — Bundles  of 
Circles  and  Clusters  of  Spheres — Pathocircles  and 
Pathospheres — One-to-One  Correlation 73—85 

LECTURE  VI 

Non-geometric  Interpretation 

Not  All  that  Glitters  is  Gold — A  Diamond  Test  of  Harmony 
— Two-dimensional  Doctrine  of  Number  Dyads  and 
Systems  Thereof — The  Three  Dimensional  Analogue. . .     86-103 

LECTURE  VII 

Essential  Discriminations 

Distinction  of  Doctrine  and  Method — Analytic  Geometry 
and  Geometric  Analysis — The  Twain  Begotten  of  Con- 
verse Transformations — An  Infinite  Family  of  Sisters — 


CONTENTS  xl 

PAGES 

All  Heritors  of  Their  Mother's  Form — Their  Common 
Character  aud  Individualities — Excessive  Meaning  of 
Content — Generic  and  Specific  Meanings  of  Euclidean 
and  Non-Euclidean — Three  Properties  Common  to 
Postulate  Systems — Fertility  and  Compendence  and 
Compatibility 104-1 1 7 

LECTURE  VIII 

Postulate  Properties 

Scientific  Platforms — Their  Fertility,  Compendence  and 
Compatibility — Differences  of  Equivalent  Platforms — 
Varieties  of  Platforms  and  Functions — Meanings  of 
Independence  and  Categoricalness — Theoretical  and 
Practical  Doubt 1 18-129 

LECTURE  IX  f 

Truth  and  the  Critic's  Art 

Mathematical  Philosophy  in  the  Role  of  Critic — A  World 
Uncriticised,  the  Garden  of  the  Devil — "Supersimian" 
Wisdom — Autonomous  Truth  and  Autonomous  False- 
hood— Other  Varieties  of  Truth  and  Untruth — Mathe- 
matics the  Study  of  Fate  and  Freedom — Its  Pure 
Branches  as  Doctrinal  Functions — Its  Applied  Branches 
as  Doctrines — The  Prototype  of  Reasoned  Discourse 
Often  Disguised  as  in  The  Declaration  of  Independence, 
The  Constitution  of  the  United  States,  The  Origin  of 
Species,  The  Sermon  on  the  Mount 130-152 

LECTURE  X 

Transformation 

Nature  of  Mathematical  Transformation — No  Transforma- 
tion, No  Thinking — Transformation  Law  Essentially 
Psychological — Relation  and  Function  and  Transforma- 
tion  as   Three   Aspects   of  One  Thing — Its    Study   the 


xii  CONTENTS 

PAGES 

Common  Enterprise  of  Science — The  Art  of  Mathe- 
matical Rhetoric — The  Static  and  the  Dynamic  Worlds 
— The  Problem  of  Time  and  Kindred  Problems — Impor- 
tation of  Time  and  Suppression  of  Time  as  the  Classic 
Devices  of  Sciences 153-180 

LECTURE  XI 
Invariance 

The  Ages-old  Problem  of  Permanence  and  Change — The 
Quest  of  What  Abides  in  a  Fluctuant  World — The 
Binding  Thread  of  Human  History — The  Tie  of  Com- 
radeship Among  the  Enterprises  of  the  Human  Spirit — 
Need  of  Critical  History  of  Thought 181-199 

LECTURE  XII 

The  Group  Concept 

The  Notion  Simply  Exemplified  in  Many  Fields — Is  "Mind" 
a  Group? — Groups  as  Instruments  for  Delimiting  Doc- 
trines —  Connection  of  Group  with  Transformation  and 
Invariance — The  Idea  Foreshadowed  in  the  Ages  of 
Speculation — The  Philosophy  of  the  Cosmic  Year — The 
Idea  of  Progress 200-232 

LECTURE  XIII 

Variables  and  Limits 

A  Glance  at  the  Shadowy  Background  of  Scientific  Ideas — The 
Meanings  of  Variable  and  Constant — Ranges  of  Variation 
and  the  Idea  of  Neighborhood — Various  Definitions  of 
Limit  Clarified  by  Simple  Examples — The  Scandal  of  a 
Starving  Nurse  in  the  Richest  Land  Known  to  History.  .   233-264 

LECTURE  XIV 

More  about  Limits 

Further  Definitions  of  Limit — Limits  and  the  Infinitesimal 
Calculus — Connection  with  Order,  Series  and  Sequence 


CONTENTS  xiii 

PAGES 

— Limits  and  Limit  Processes  Omnipresent  as  Ideals 
and  Idealization  in  All  Thought  and  Human  Aspiration 
— Ideals  the  Flint  of  Reality — Genius  and  General- 
ization     265-296 

LECTURE  XV 

Infinity 

Mathematical  Infinity — Its  Dynamic  and  Static  Aspects — 
Need  of  History  of  the  Imperious  Concept — The  Role 
of  Infinity  in  a  Mighty  Poem — No  Infinity,  No  Science.  .   297-316 

LECTURE  XVI 

Hyperspaces 

Meaning  of  Dimensionality — Spaces  of  Four  or  More  Dimen- 
sions— The    Mode    of  Their    Existence — Distinction    of 
Imagination    and    Conception — Logical  Existence    and 
Sensuous    Existence — Open    Avenues    to    Unimaginable 
Worlds 317-341 

LECTURE  XVII 

NON-EUCLIDEAN   GEOMETRIES 

Their  Birth  and  Varieties — Their  Logical  Perfection — Their 
Psychological  Differences — Their  Scientific  and  Philo- 
sophic Significance — All  of  Them  Pragmatically  True — 
Science  and  Tragedy — A  Prelude  on  the  Popularization 
of  Science — Science  and  Democracy 342-365 

LECTURE  XVIII 

The  Mathematics  of  Psychology 

Backwardness  of  This  and  of  the  Psychology  of  Mathematics 
— The  Law  of  Weber  and  Fechner  Reexamined — Some  of 
the  Law's  Unnoticed  Implications — Thought  as  Infinitely 
Refined  Sensibility 366-401 


xiv  CONTENTS 

PAGES 

LECTURE  XIX 

The  Psychology  of  Mathematics 

Retardation  of  Mathematics  and  Science  by  Backward  Psy- 
chology— Psychology  of  Mathematics  Essential  to  Best 
Mathematical  Teaching — Questions  for  Psychologists — 
Symmetry  of  Thought  and  Asymmetry  of  Imagination. .   402-421 

LECTURE  XX 

Korzybski's  Concept  of  Man 

What  Time-binding  Means — Dimensionality  and  the  Math- 
ematical Theory  of  Logical  Types — The  Natural  Law  of 
Civilization  as  an  Increasing  Exponential  Function  of 
Time — Human  Ethics  as  Time-binding  Ethics,  Not  the 
Space-binding  Ethics  of  Animals 422-451 

LECTURE  XXI 

Science  and  Engineering 

Change  of  Emphasis  from  Non-human  to  Human  Energies — 
Science  as  Engineering  in  Preparation — Engineering  as 
Science  in  Action — Mathematics  the  Guide  of  the 
Engineer — Engineering  the  Guide  of  Humanity — 
Humanity  the  Civilizing  or  Time-binding  Class  of 
Life — The  Four  Defining  Marks  of  the  Great  Engineer 
of  the  Future — Engineering  Statesmanship 452-462 


MATHEMATICAL  PHILOSOPHY 


MATHEMATICAL    PHILOSOPHY 


LECTURE  I 

Introduction 

INTELLECTUAL  FREEDOM  AND  LOGICAL  FATE MATHE- 
MATICAL OBLIGATIONS  OF  PHILOSOPHY  AND  EDU- 
CATION  COMMON    HUMANITY   AND    INDIVIDUALITY 

HUMANISTIC  AND  INDUSTRIAL  EDUCATION MAN 

NOT   AN    ANIMAL ETHICS    NOT    A    BRANCH    OF    ZO- 
OLOGY  EXCELLENCE  AND  THE  MUSES LOGIC  THE 

MUSE    OF    THOUGHT THE     HEROIC    TRADITION     IN 

PHILOSOPHY RADIANT  ASPECTS  OF  AN  OVER-WORLD 

It  is  the  aim  of  the  following  lectures  to  point  out, 
in  a  manner  suitable  for  you  as  students  of  Thought,  and 
to  submit  for  your  consideration,  some  of  the  more  es- 
sential and  more  significant  relations  between  Mathe- 
matics and  Philosophy.  Each  of  these  great  terms  is  to 
be  understood  in  its  most  embracing  sense.  Mathema- 
ticians sometimes  speak  contemptuously  of  philosophy; 
and  philosophers  sometimes  speak  contemptuously  of 
mathematics.  The  contempt  thus  manifested  does  not 
spring  from  mathematics  in  the  former  case,  nor  from 
philosophy  in  the  latter;  in  both  cases  it  springs  out  of 
ignorance — philosophical    ignorance    of    mathematicians 


3  MATHEMATICAL   PHILOSOPHY 

and  mathematical  ignorance  of  philosophers.  No  doubt 
philosophically  unenlightened  mathematicians  and  mathe- 
matically unenlightened  philosophers  will  quarrel  in  the 
future  as  in  the  past;  but  in  the  future  as  in  the  past, 
the  quarreling  and  the  sneering  will  be  the  quarreling  and 
sneering  of  men  and  not  of  the  great  subjects  they  rep- 
resent and  misrepresent;  for  between  the  spirit  of  mathe- 
matics and  the  spirit  of  philosophy  there  is  no  discord,  no 
antagonism,  no  strife;  they  are  by  their  natures  friendly 
rivals  in  the  pursuit  of  truth  and  light;  they  are  compan- 
ions in  excellence;  they  are  comrades  in  the  service  of 
wisdom. 

I  have  said  that  the  "aim"  of  these  lectures  is  to  dis- 
close fundamental  connections  between  mathematics  and 
philosophy.  What  I  have  described  as  their  "aim"  is 
not  so  much  the  aim,  or  end,  as  a  means.  For  it  will  be- 
come increasingly  evident  as  we  advance  that  the  work 
we  are  to  be  engaged  in  is  fundamentally  the  study  of 
Fate  and  Freedom — logical  fate  and  intellectual  freedom. 
I  mention  the  matter  here  because  you  ought  to  have  it 
consciously  in  mind  from  the  beginning.  You  should  bear 
it  in  mind  at  every  stage  of  the  discussion,  even  in  con- 
nections where  so  warm  an  interest  may  seem  remote.  A 
preliminary  word  of  explanation  is  therefore  desirable. 

We  are  going  to  deal  with  ideas — with  their  charac- 
ters, with  their  meanings,  with  their  relations.  Now,  an 
idea  is  in  Itself  an  eternal  thing  and  the  relations  of  an 
idea  with  other  ideas  are  eternal.  An  idea  is  just  what 
it  is  and  it  is  unalterable;  a  relation  among  ideas  is  just 
what  it  is  and  it  is  unalterable.  We  do,  indeed,  often 
speak  as  if  such  were  not  the  case;  we  habitually  speak 
as  if  ideas  and  their  relations  were  temporal  affairs,  im- 
permanent,  mutable,   malleable,   capable   of   growth,    of 


INTRODUCTION  S. 

modification,  of  decay,  of  destruction,  as  when  we  say, 
for  example,  that  we  have  "changed"  our  ideas  or  that 
such-and-such  an  idea  has  "grown"  in  importance  or  has 
"become"  sterile  or  is  "dead."  It  is,  I  fancy,  hardly 
necessary  to  say  that  all  such  ways  of  speaking  are  figura- 
tively-convenient no  doubt,  often  pleasing,  sometimes 
very  effective,  yet  thoroughly  figurative, — and  that,  if 
taken  literally,  they  quickly  and  inevitably  lead  to  scien- 
tific and  philosophic  disaster.  You  or  I  may  abandon  an 
idea  that  we  have  held  and  we  may  adopt  an  idea  that  is 
new  to  us;  the  "old"  one  and  the  "new"  one  may  closely 
resemble  each  other;  they  may  indeed  be  identical  in  some 
respect  and  may  even  be  called  by  the  same  name;  but 
neither  of  them  has  been  transmuted  into  the  other;  each 
of  them  remains  and  will  remain  just  what  it  was.  Let 
me  illustrate  the  eternality  of  ideas  and  of  their  relations 
by  means  of  a  simple  example.  You  know  that  in  dis- 
course ideas  are  represented  by  symbols — by  words  or 
other  signs.  Consider  the  symbols  2,  7,  9,  +,  and  = ; 
each  of  them  stands  for  an  idea  familiar  to  all  of  us. 
The  symbols  are  man-made;  but  the  things  they  stand 
for,  though  they  were  discovered  by  man,  are  not  man- 
made;  they  are  increate,  as  Milton  would  say,  and  inde- 
structible, and  the  like  is  true  of  their  relations;  one  of 
these  is  expressed  by  the  statement  (a)  2+7=9;  the 
statement  expressing  the  relation  is  a  creature  of  man,  but 
the  relation  itself  is  not — man  discovered  it,  but  he  did 
not  make  it — it  is  a  thing  increate  and  indestructible,  the 
same  yesterday,  today  and  forever.  The  truth  of  what 
I  have  just  now  said  is  very  evident,  but  the  illustration 
is  arithmetical.  Is  the  eternality  equally  evident  in  the 
case  of  all  other  ideas  and  their  relations?  No,  it  is  not 
equally  evident,  but  it  is  none  the  less  true.     Shall  we  take 


4  MATHEMATICAL   PHILOSOPHY 

another  example?  Let  us  take  one  that  is  very  far 
from  being  specifically  arithmetical.  Consider  the  state- 
ment : 

(b)  If  something  S  has  the  property  p  and  whatever 
has  the  property  p  has  the  property  />',  then  S  has  the 
property  p'. 

You  observe  that  the  statement  expresses  a  certain 
relation  among  certain  ideas — the  idea,  for  example, 
denoted  by  "something,"  that  denoted  by  "property"  (or 
quality  or  mark),  that  denoted  by  "whatever,"  and  so 
on.  The  denoting  terms  are  indeed  man-made,  but  the 
ideas  denoted  are  not,  they  are  merely  man-discovered 
and  man-known;  and  the  statement  expressing  the  relation 
is  a  creature  of  man,  but  the  relation  itself,  though  man 
discovered  it,  was  not  created  by  him :  it  is  an  unoriginated 
thing,  immutable,  universal,  timeless.  The  illustration  is 
very  general,  very  abstract  and  very  cold.  Perhaps  you 
prefer  something  warmer,  more  specific,  more  concrete. 
Well,  it  is  easy  to  find  such,  for  the  foregoing  general 
statement  is  infinitely  rich  in  concrete  applications.  Let 
me  instance  one  of  them,  one  that  is  sufficiently  warm; 
it  is  indeed  one  that  goes  to  the  very  heart  of  our  human 
ethics — not  to  our  ethics  as  it  is,  but  as  it  ought  to  be  and 
as  no  doubt  it  will  be.    The  application  is  this,  namely: 

If  human  beings  are  by  nature  civilization-builders, 
or  "time-binders,"  and  if  all  time-binders,  or  civiliza- 
tion-builders, are  both  inheritors  from  the  toil  of  by- 
gone generations  and  trustees  for  the  generations  to 
come,  then  we  humans  stand  in  the  double  relationship 
— debtors  of  the  dead,  trustees  of  the  unborn — thus 
uniting  past,  present  and  future  in  one  living,  growing 
reality. 


INTRODUCTION  5 

The  infinite  and  eternal  significance  of  that  fact  may, 
I  trust,  be  left  for  your  meditation. 

Without  more  talk  and  without  danger  of  misunder- 
standing, we  may,  I  believe,  now  speak  of  ideas  as  con- 
stituting a  world — the  world  of  ideas.  With  that  world 
all  human  beings  as  human  have  to  deal — there  is  no 
escape;  it  is  there  and  only  there  that  foundations  are 
found — foundations  for  science,  foundations  for  philoso- 
phy, foundations  for  art,  foundations  for  religion,  for 
ethics,  for  government  and  education;  it  is  in  the  world 
of  ideas  and  only  there  that  human  beings  as  human  may 
find  principles  or  bases  for  rational  theories  and  rational 
conduct  of  life,  whether  individual  life  or  community  life; 
choices  differ  but  some  choice  of  principles  we  must  make 
if  we  are  to  be  really  human — if,  that  is,  we  are  to  be 
rational — and  when  we  have  made  it,  we  are  at  once 
bound  by  a  destiny  of  consequences  beyond  the  power  of 
passion  or  will  to  control  or  modify;  another  choice  of 
principles  is  but  the  election  of  another  destiny.  The 
world  of  ideas  is,  you  see,  the  empire  of  Fate. 

Is  the  human  Intellect,  then,  a  slave?  No:  it  is  free; 
but  its  freedom  is  not  absolute;  it  is  limited  by  fact  and 
by  law — by  the  laws  of  thought,  by  the  immutable  char- 
acters of  ideas  and  by  their  unchanging  eternal  relation- 
ships. Intellectual  freedom  is  freedom  to  think  in  accord 
with  the  laws  of  thought,  in  accord  with  the  natures  of 
ideas,  in  accord  with  their  interrelations,  which  are  un- 
alterable. And  no  variety  of  human  freedom — no  insti- 
tution erected  in  its  sacred  name — if  it  does  not  conform 
to  the  eternal  conditions  of  intellectual  freedom — can 
stand. 

What  I  have  now  said  is,  I  hope,  a  sufficient  prelimi- 
nary intimation  of  what  I  mean  by  saying  that  our  work 


6  MATHEMATICAL   PHILOSOPHY 

in  these  lectures  is  to  be  fundamentally  a  study  of  Free- 
dom and  Fate. 

Your  major  interest  is  in  philosophy;  mine  is  in  mathe- 
matics. You  have  besides,  I  trust,  a  lively,  if  only  a 
minor,  interest  in  mathematics,  as  I  have  had  from  the 
days  of  my  youth  a  genuine  interest,  albeit  a  subordinate 
one,  in  the  concerns  of  philosophy  and  especially  in  the 
philosophy  of  mathematics.  It  is,  I  believe,  a  happy  cir- 
cumstance that  your  interest  and  mine  in  these  great  sub- 
jects are  thus  complementary  instead  of  coincident  or 
antagonistic;  for  in  this  relation  of  interest  there  is  im- 
plied a  corresponding  relation  of  attainment,  limitation, 
outlook  and  temper;  and  this  relation,  if  wc  bear  it  in 
mind,  will  be  favorable  in  important  ways  to  the  pros- 
perity of  our  enterprise;  for  example,  it  should,  on  the 
one  hand,  have  the  effect  of  restraining  me  from  adduc- 
ing too  lightly  or  too  freely,  with  too  little  explanation, 
mathematical  considerations  with  which  you  may  justly 
feel  I  have  no  right  to  suppose  you  familiar;  and,  on  the 
other  hand,  when  you  discover,  as  you  will  doubtless  fre- 
quently discover,  that  I  have  fallen  into  error  because  of 
my  philosophical  limitations,  it  will,  I  hope,  make  you 
feel  it  your  duty  to  "impose  upon  me  the  just  retribution," 
in  accordance  with  the  saying  of  Plato  that  "The  just 
retribution  of  him  who  errs  is  that  he  be  set  right." 

It  need  hardly  be  said  that  no  one  should  follow  this 
course  in  the  hope  of  thereby  acquiring  mathematical 
knowledge  or  skill  in  the  usual  sense  of  these  terms.  I 
assume  that  what  is  mainly  responsible  for  your  presence 
here  is  a  desire  and  a  hope  of  a  different  kind:  you  desire 
to  gain  insight  into  the  essential' nature  of  mathematics 
regarded  as  a  distinctive  type  of  thought;  you  desire  to 
acquire  knowledge  of  what  is  characteristic  and  funda- 


INTRODUCTION  7 

mental  In  mathematical  method;  you  hope  to  gain  ac- 
quaintance with  some  of  the  great  mathematical  concepts, 
with  such  of  the  dominant  concepts  as  are  accessible  to 
laymen;  you  desire  to  win  a  just  sense  of  the  spiritual 
significance  of  the  science;  in  a  word,  your  quest  is  for 
such  an  understanding  of  it  as  will  help  you  to  view 
mathematics  in  a  vast  perspective — in  relation,  that  is, 
with  the  other  sciences  and  arts  and  the  other  modes  and 
forms  of  human  activity.  Such,  I  take  it,  are  the  ends 
that  define  our  task.  I  should  indeed  be  unhappy  if  I 
did  not  hope  that  the  lectures,  though  they  have  been 
fashioned  with  controlling  reference  to  the  task  indicated, 
will  at  the  same  time  serve  in  some  measure  to  extend 
your  acquaintance  with  the  existing  body  of  mathematical 
doctrine.  But  it  is  to  be  understood  that  this  result,  if 
the  lectures  produce  it,  will  be  incidental  and  subsidiary 
to  their  main  purpose;  for  they  are  not  designed  to  teach 
a  recognized  branch  of  mathematics  whether  elementary 
or  more  advanced.  Mathematical  students  having  little 
or  no  interest  in  the  philosophy  of  their  science  must  be 
frankly  counseled  to  repair  to  other  courses  for  the  kind 
of  instruction  they  desire.  And  students  of  philosophy 
should  not  indulge  themselves  in  the  vain  hope  of  acquir- 
ing mathematical  knowledge  by  merely  "philosophizing" 
about  the  subject  or  by  pensively  gazing  upon  its  general 
aspects  from  an  external  point  of  view.  From  time  im- 
memorial, there  has  been  but  one  way  to  become  a  mathe- 
matician and  there  will  never  be  another:  it  is  a  way 
interior  to  the  subject  and  involves  years  of  assiduous 
toil.  Short  cuts  to  mathematical  scholarship  there  is 
none,  whether  the  seeker  be  a  philosopher  or  a  king. 

How  much  mathematical  training  is  essential  to  the 
qualification  of  one  who  may  hope  to  follow  the  lectures 


8  MATHEMATICAL   PHILOSOPHY 

profitably?  It  is  natural  that  you  should  wish  to  ask  that 
question  at  this  point.  The  question  is  important  and 
the  answer  easy  and  short:  so  much  mathematical  train- 
ing— so  much  knowledge  of  algebra,  geometry  and  trigo- 
nometry— as  a  capable  student  can  acquire  in  one  col- 
legiate year.  Compared  with  the  existing  science  of 
mathematics  such  knowledge  is  very  meagre,  a  bare  be- 
ginning; but,  taken  absolutely,  it  is  much;  in  respect  of 
content  or  mere  information  as  distinguished  from  insight 
and  power,  it  is  far  more  than  Thales  had,  or  Pythagoras 
or  even  Plato  or  even  Galileo.  It  would  be  very  con- 
venient if  I  might  assume  more;  projective  geometry,  for 
example,  and  some  acquaintance  with  analytical  geometry 
— which  should  remind  you  of  Descartes,  and  with  the 
calculus — which  should  remind  you  of  Leibniz ;  for  I  shall 
be  obliged  occasionally  to  employ  ideas  drawn  from  these 
and  other  branches  of  mathematics,  and  shall  have  to 
interrupt  and  delay  the  discussion  a  little  in  order  to  ex- 
plain the  ideas  as  the  necessity  arises  for  using  them. 
Perhaps  I  should  add  that,  for  understanding  the  lectures, 
a  certain  intellectual  maturity,  logical  acumen,  open-mind- 
edness  and  philosophical  insight  are  not  less  essential  than 
the  stated  minimum  of  mathematical  knowledge. 

I  desire  to  invite  you  now  to  a  somewhat  compre- 
hensive consideration  of  a  much  larger  question,  one  of 
greater  difficulty  and  far  greater  importance — a  question 
of  both  general  and  permanent  interest.  The  question 
is:  How  much  mathematical  training — how  much  mathe- 
matical knowledge,  discipline,  and  habit — may  be  reas- 
onably regarded  as  indispensable  to  the  proper  equip- 
ment of  a  philosopher?  It  may  well  be  that  you  will  be 
qualified  to  give  a  better  answer  at  the  end  of  the  course 
than  that  which  I  am  about  to  submit  here  at  the  begin- 


INTRODUCTION  0 

ning  of  it.  Nevertheless,  I  am  disposed  to  think  that  a 
preliminary  discussion  of  the  matter  will  be  of  some  serv- 
ice. A  complete  discussion  would  involve  many  consid- 
erations differing  greatly  in  weight.  I  shall  ask  your 
attention  to  such  of  them  as  seem  to  me  cardinal  and 
decisive. 

The  first  consideration  grows  out  of  the  fact  that  a 
philosopher  is  a  human  being.  It  is  immediately  evident 
that  the  proper  equipment  of  a  philosopher  must  include 
as  much  mathematical  training  as  is  essential  to  the  ap- 
propriate education  of  men  and  women  as  human  beings. 
How  much  is  that?  Be  good  enough  to  note  what  the 
question  precisely  is.  I  am  not  asking  how  much  mathe- 
matical discipline  is  essential  to  a  "liberal  education"  for 
this  fine  term,  though  clearly  defined  long  ago  by  Aris- 
totle in  terms  of  spiritual  interest  and  attitude,  has  in 
our  day  lost  its  significance  even  for  the  majority  of  aca- 
demic folk,  who  ought  to  be  ashamed  of  the  fact.  That 
great  man,  the  late  Lord  Kelvin,  used  to  tell  his  students 
that  among  the  "essentials  of  a  liberal  education  is 
mastery  of  Newton's  Principia  and  Herschel's  As- 
tronomy." On  the  other  hand,  such  educators  as  Mat- 
thew Arnold,  John  Henry  Newman,  Thomas  Huxley, 
though  differing  infinitely  in  their  outlooks  upon  the 
world  and  in  their  estimates  of  worth,  yet  unite  in  deny- 
ing Kelvin's  contention  impetuously  or  even  with  scorn. 
Let  us  so  frame  our  question  as  to  avoid  that  debate.  The 
question  is:  How  much  mathematical  discipline  is  es- 
sential to  the  appropriate  education  of  men  and  women 
as  human  beings?  This  exceedingly  important  question 
admits  of  a  definite  answer  and  it  admits  of  it  in  terms  of 
a  supremely  important  and  incontestable  general  prin- 
ciple.    A  clue  to  the  principle  is  found  in  the  phrase  I 


10  MATHEMATICAL   PHILOSOPHY 

have  just  now  employed:  education  of  men  and  women 
as  human  beings.  Before  stating  the  principle,  it  will 
be  convenient  to  give  it  a  name.  I  shall  call  it  the  Prin- 
ciple of  Humanistic  Education  as  distinguished  from 
what  has  come  to  be  designated  in  our  day  as  Industrial 
Education.  I  say  "as  distinguished  from"  because  the 
two  varieties  of  education,  whether  they  be  compared 
with  respect  to  the  conceptions  which  lie  at  the  heart  of 
them  or  with  respect  to  the  motives  which  actuate  and 
sustain  them,  are  widely  different.  In  order  to  set  the 
principle  in  a  clear  light,  let  me  indicate  briefly  the  obvi- 
ous facts  lying  at  its  base  and  leading  naturally  to  its 
formulation. 

What  the  individuals  composing  our  race  have  in 
common  falls  into  two  parts:  a  part  consisting  of  those 
numerous  instincts,  impulses,  traits,  propensities  and 
powers  which  we  humans  have  in  common,  not  only  with 
one  another,  but  with  many  of  the  creatures  constituting 
the  world  of  animals — a  subhuman  *  world;  and  a  second 
part  consisting  of  such  instincts,  impulses,  traits,  propen- 
sities and  powers  as  are  distinctively  human.  These 
latter,  we  may  say,  constitute  our  Common  Humanity. 
They  present,  indeed,  an  endless  variety  of  detail,  but  in 
the  long  course  of  man's  experience  with  man  he  has 
learned  to  group  them,  in  accordance  with  their  principal 
aspects,  into  a  small  number  of  familiar  classes.  .  And 
accordingly,  the  nature  of  our  common  humanity  is  fairly 
well  characterized  by  saying  that  human  beings  as  such 
possess  in  some  recognizable  measure  such  marks  as  the 
following:  a  sense  for  language,  for  expression  in  speech 
— the  literary  faculty;  a  sense  for  the  past,  for  the  value 

1  See  Lecture  XX  for  a   discussion  of  Korzybski's  concept  of  Man  in 
terms  of  Time — a  conception  according  to  which  humans  are  not  animals. 


INTRODUCTION  11 

of  experience — the  historical  faculty;  a  sense  for  the 
future,  for  prediction,  for  natural  law — the  scientific 
faculty;  a  sense  for  fellowship,  cooperation,  and  justice 
— the  political  faculty;  a  sense  for  the  beautiful — the 
artistic  faculty;  a  sense  for  logic,  for  rigorous  thinking 
— the  mathematical  faculty;  a  sense  for  wisdom,  for 
world  harmony,  for  cosmic  understanding — the  philo- 
sophical faculty;  and  a  sense  for  the  mystery  of  divinity 
— the  religious  faculty. 

Such  are  the  evident  tokens  and  the  cardinal  constitu- 
ents of  that  which  in  human  beings  is  human.  It  is  es- 
sential to  note  that  to  each  of  the  senses  or  faculties  in 
virtue  of  which  humans  are,  not  animals,  but  a  higher 
class  of  beings,  there  corresponds  a  certain  type  of  dis- 
tinctively human  activity — a  kind  of  activity  in  which  all 
human  beings,  whatever  their  stations  or  occupations,  are 
as  humans  obliged  to  participate.  Like  the  faculties  to 
which  they  correspond,  these  types  of  activity,  though 
they  are  interrelated,  are  yet  distinct.  Each  of  them  has 
a  character  of  its  own.  Above  each  of  the  types  there 
hovers  a  guardian  angel — an  ideal  of  excellence — wooing 
our  loyalty  with  a  benignant  influence  superior  to  every 
compulsive  force  and  every  authority  that  may  command. 
Nothing  more  precious  can  enter  a  human  life  than  a 
vision  of  these  angels,  and  it  is  the  revealing  of  them 
that  humanistic  education  has  for  its  function  and  its  aim. 
Stated  in  abstract  terms  the  principle  is  this:  Each  of 
the  great  types  of  distinctively  human  activity  owns  an 
appropriate  standard  of  excellence;  it  is  the  aim  of  hu- 
manistic education  to  lead  the  student  into  a  clear  knowl- 
edge of  these  standards  and  to  give  him  a  vivid  and 
abiding  sense  of  their  authority  in  the  conduct  of  life. 
Ethics  is  not  a  branch  of  Zoology. 


12  MATHEMATICAL    PHILOSOPHY 

It  is  plain  that  this  conception  stands  in  sharp  con- 
trast with  the  central  idea  of  industrial  education.  For 
humanistic  education  has  for  its  aim,  as  I  have  said,  the 
attainment  of  excellence  in  the  things  which  constitute 
our  common  humanity.  On  the  other  hand,  industrial 
education  is  directly  and  primarily  concerned  with  our 
individualities.  It  might,  therefore,  be  more  appropri- 
ately called  individualistic  education.  It  regards  the 
world  as  an  immense  camp  of  industries  where  endlessly 
diversified  occupations  call  for  special  propensities,  gifts, 
and  training.  Accordingly  its  aim,  its  ideal,  is  to  detect 
in  each  youth  as  early  as  may  be  the  presence  of  such 
gifts  and  propensities  as  tend  to  indicate  and  to  qualify 
him  for  some  specific  form  of  calling  or  bread-winning 
craft;  then  to  counsel  and  guide  him  in  the  direction 
thereof;  and  finally,  by  way  of  education,  to  teach  him 
those  things  which,  in  the  honorable  sense  of  the  phrase, 
constitute  the  tricks  of  the  trade. 

What  are  we  to  say  of  it?  The  answer  is  obvious. 
Industrial  education,  rightly  conceived,  is  essentially  com- 
patible with  the  humanistic  type;  it  may  breathe  the 
humanistic  spirit;  the  two  varieties  of  education  are  essen- 
tial to  constitute  an  ideal  whole,  for  human  beings  possess 
both  individuality  and  the  common  humanity  of  man.  In- 
dustrial education,  when  thus  regarded  as  supplementary 
to  humanistic  education,  is  highly  commendable ;  but  when 
it  is  viewed  as  an  equivalent  for  the  latter  or  as  an  ideal 
substitute  for  it,  it  is  ridiculous,  contemptible  and  vicious. 
For  the  fact  must  not  be  concealed  that  a  species  of  edu- 
cation which,  in  producing  the  craftsman,  neglects  the 
man,  is,  in  point  of  kind  and  principle,  precisely  on  a  level 
with  that  sort  of  training  which  teaches  the  monkey  and 


INTRODUCTION  13 

the  bear  to  ride  a  bicycle,  or  the  seal  to  balance  a  staff 
upon  its  nose  or  to  twirl  a  disc. 

These  considerations  are  no  doubt  obvious.  I  should 
not  dwell  upon  them  at  so  great  length  but  for  the  fact 
that  in  the  excitement  and  confusion  of  our  industrial 
age  the  most  obvious  of  important  facts  and  the  most 
evident  of  important  principles  are  so  commonly  lost 
sight  of  that  they  require  to  be  cited  again  and  again 
and  again.  Nowhere  is  the  confusion  of  the  time  more 
evident  than  in  the  somewhat  noisy  and  sometimes  acri- 
monious discussion  that  has  been  recently  and  still  is  go- 
ing on  throughout  our  country  regarding  the  value  of 
mathematics  as  a  subject  in  secondary  and  collegiate  edu- 
cation. The  instigators  of  the  discussion,  those,  that  is, 
who  advocate  so  reducing  mathematical  requirements  as 
practically  to  abolish  the  subject  from  curricula  of  general 
education,  are  not  malicious  nor  insincere;  many  of  them, 
I  do  not  doubt,  are  well-meaning  citizens.  And  if  their 
rather  voluminous  discourses  are  often  singularly  lacking 
in  coherence,  in  clarity  and  in  depth,  the  defects  are  not 
due  to  evil  intentions  but  rather,  I  suspect,  to  confusion 
and  a  lack  of  just  that  sort  of  discipline  which  the  subject 
the  authors  are  engaged  in  depreciating  is  peculiarly 
qualified  to  give.  Perhaps  we  should  not  be  astonished. 
If  the  saying  of  Sir  Oliver  Lodge  be  true  that  "the  mathe- 
matical ignorance  of  the  average  educated  person  has 
always  been  complete  and  shameless,"  one  ought  not,  I 
suppose,  to  be  too  much  astonished  if  in  a  vast,  crude, 
formless,  sprawling  democracy  like  ours,  a  way  to  edu- 
cational "leadership"  is  sometimes  found  by  men  whose 
innocence,  not  only  of  mathematics  but  of  the  other  great 
subjects,  including  the  principles  of  education,  is  well- 
nigh  complete  and  shameless.     And  yet,  despite  famil- 


14  MATHEMATICAL   PHILOSOPHY 

iarity  with  the  phenomenon,  it  is  sometimes  a  bit  hard  to 
avoid  astonishment  and  even  a  loss  of  patience.  Not 
long  ago  a  high-placed  counselor  of  a  well-known  college 
of  liberal  arts  challenged  me,  with  defiant  confidence  and 
unfeigned  solemnity,  to  give  any  good  reason  why  college 
students  should  be  required  to  pursue  a  course  in  algebra 
rather  than  one  in  some  practical  art,  say  the  art  of  cook- 
ing mutton  chops.  On  receiving  such  a  challenge  from 
a  grown  man,  what  should  a  grown  man  do?  Confess 
his  astonishment?  Betray  an  exhaustion  of  patience? 
Fly  to  the  easy  refuge  of  ridicule?  Any  such  reaction 
would  probably  have  been  misunderstood.  In  dealing 
with  a  solemn  question,  no  matter  how  stupid,  it  is  usually 
the  wiser  course  to  treat  it  with  respect  if  possible.  I 
might  have  responded,  in  the  fine  words  of  Professor 
Whitehead,1  that 

"Algebra  is  the  intellectual  instrument  which  has 
been  created  for  rendering  clear  the  quantitative  as- 
pects of  the  world.  .  .  .  Through  and  through  the 
world  is  infected  with  quantity.  To  talk  sense,  is  to 
talk  in  quantities.  It  is  no  use  saying  that  the  nation 
is  large, — How  large?  It  is  no  use  saying  that  radium 
is  scarce, — How  scarce?  You  can  not  evade  quantity. 
You  may  fly  to  poetry  and  to  music,  and  quantity  and 
number  will  face  you  in  your  rhythms  and  your  octaves. 
Elegant  intellects  which  despise  the  theory  of  quantity 
are  but  half  developed.  They  are  more  to  be  pitied 
than  blamed." 

It  did  not  seem  to  me,  however,  that  one  capable  of 
issuing  such  a  challenge  as  that  to  which  I  have  alluded 

1  A.  N.  Whitehead :  The  Organization  of  Thought.     Cambridge  Uni- 
versity Press. 


INTRODUCTION  15 

could  feel  the  weight  of  such  a  response,  and  I  did  not 
make  it.  It  is,  you  observe,  a  response  in  terms  of  quan- 
tity. Quantity  is  indeed  omnipresent  in  our  world;  but 
so,  too,  is  quality,  and  of  the  two  things,  the  latter  is  per- 
haps the  more  universal  in  its  appeal.  Algebra  is  indeed 
essential  to  the  theory  of  quantity  and  the  theory  of 
quantity  is  essential  to  the  subjugation  of  natural  resources 
to  the  use  of  man;  of  quality,  on  the  other  hand,  algebra 
is  not  a  science  but,  though  it  is  not  a  science  of  quality, 
it  has  a  quality,  a  human  quality,  to  which  it  owes  its  high 
rank  in  the  spiritual  hierarchy  of  human  disciplines.  And 
so  I  endeavored,  with  poor  success  I  fear,  to  answer  the 
challenge  in  terms  of  quality.  I  invoked  the  principle 
which  in  this  lecture  I  have  been  calling  the  principle  of 
humanistic  education.  I  sought,  that  is,  to  make  it  clear 
that,  in  contrast  with  the  practical  arts,  the  science  of  al- 
gebra as  a  discipline  possesses  a  certain  quality  by  virtue 
of  which,  if  the  subject  be  rightly  administered,  the 
student  is  gradually  brought  into  the  presence  of  one  of 
those  great  standards  of  excellence  by  which,  as  we  have 
seen,  distinctively  human  activity  in  all  its  principal  types 
is  to  be  guided  and  judged.  The  standard  to  which  I 
refer  is,  as  you  have  doubtless  surmised,  the  standard  of 
excellence  in  the  quality  of  thinking  as  thinking — the 
standard  which  mathematicians  are  accustomed  to  call 
Logical  Rigor — clarity,  that  is,  precision  and  coherence. 
And  now  the  mention  of  that  great  term  may  serve 
to  reassure  you,  should  you  have  begun  to  suspect  that 
in  the  course  of  this  rather  long  excursion  I  may  have 
forgotten  the  question  initiating  it.  The  question  is: 
How  much  mathematical  training  is  essential  to  the  ap- 
propriate education  of  men  and  women  as  human  beings? 
I  have  said  that  the  question  admits  of  a  definite  answer 


16  MATHEMATICAL   PHILOSOPHY 

in  terms  of  a  supreme  and  incontestable  principle.  I 
have  stated  the  principle  as  well  as  I  can  and  have  tried 
to  signalize  its  importance  for  a  general  theory  of  edu- 
cation. It  remains  to  apply  it  to  the  specific  question  be- 
fore us.  The  task  is  not  difficult.  It  is  plain  that  one 
of  the  great  types  of  distinctively  human  activity — per- 
haps the  greatest  and  most  distinctively  human  type — is 
what  is  known  as  Thinking — the  handling  of  ideas  as 
ideas — the  formation  of  concepts,  the  combination  of  con- 
cepts into  higher  and  higher  ones,  discernment  of  the 
relations  subsisting  among  them,  embodiment  of  these  re- 
lations in  the  forms  of  judgments  or  propositions,  the 
ordering  and  use  of  these  in  the  construction  of  doctrine 
regarding  life  and  the  world — in  a  word,  the  whole  com- 
plex of  activity  involved  in  the  discourse  of  Thought.  It 
is  essential  to  the  argument  I  am  making  to  keep  steadily 
in  mind  that  this  kind  of  activity,  our  sense  for  it,  our 
faculty  for  it,  the  need  to  which  it  ministers,  the  joy  it 
gives,  and  the  obligation  it  imposes  are  part  and  parcel 
of  what  we  have  been  calling  our  common  humanity  as 
distinguished,  on  the  one  hand,  from  that  which  is  animal 
in  man,  and,  on  the  other,  from  such  special  propensities 
or  other  marks  as  give  the  differing  specimens  of  human- 
kind their  respective  individualities.  Thinking  is  not  in- 
deed essential  to  life,  but  it  is  essential  to  human  life.  All 
men  and  women  as  human  beings  are  inhabitants  of  the 
Gedankenwelt — citizens,  so  to  speak,  of  the  world  of 
ideas,  native  citizens  of  the  world  of  thought.  And  now 
what  shall  we  say  is  the  prototype  of  excellence  in  think- 
ing? What  is  the  hovering  angel  wooing  our  loyalty  to 
what  is  best  in  thinking?  What  is  the  muse  of  life  in 
the  world  of  ideas?  An  austere  goddess,  high,  pure, 
serene,  cold  towards  human  frailty,   demanding  perfect 


INTRODUCTION  17 

precision  of  ideas,  perfect  clarity  of  expression,  and  per- 
fect allegiance  to  the  eternal  laws  of  thought.  In  mathe- 
matics the  name  of  the  muse  is  familiar:  it  is  Rigor — 
Logical  Rigor,  which  signifies  a  kind  of  silent  music,  the 
still  harmony  of  ideas,  the  intellect's  dream  of  logical 
perfection. 

Can  the  dream  be  realized?  I  am  well  aware  that 
most  of  the  things  which  constitute  the  subject-matter  of 
our  human  thinking — that  most  of  the  things  to  which 
our  thought  is  drawn  by  interest  or  driven  by  the  exigen- 
cies of  life — are  naturally  so  nebulous,  so  vague,  so 
indeterminate  that  they  cannot  be  handled  in  strict 
accordance  with  the  rigorous  demands  of  logic.  I  am 
aware  that  these  demands  can  not  be  fully  satisfied  even 
in  mathematics,  the  logical  science  par  excellence.  Never- 
theless I  contend  that,  as  the  ideal  of  excellence  in  think- 
ing, Logical  Rigor  is  supremely  important,  not  only  in 
mathematical  thinking,  but  in  all  thinking  and  especially 
in  just  those  subjects  where  precision  is  least  attainable. 
For  without  this  ideal,  thinking  is  without  a  just  standard 
for  self-criticism,  and  without  light  upon  its  course;  it  is 
a  wanderer,  like  a  vessel  at  sea  without  compass  or  star. 
Were  it  necessary,  how  easy  it  would  unfortunately  be 
to  cite  endless  examples  of  such  thinking  from  the  multi- 
tudinous writings  of  our  time.  Indeed,  if  the  pretentious 
books  produced  in  these  troubled  years  by  men  without 
logical  insight  or  a  sense  of  logical  obligation  were 
gathered  into  a  heap  and  burned,  they  would  thus  produce, 
in  the  form  of  a  bright  bonfire  the  only  light  they  are 
qualified  to  give.  "Logic,"  it  has  been  said,  "is  the  child 
of  a  good  heart  and  a  clear  head."  We  know,  however, 
that  an  evil  heart  is  not  essential  to  a  fool  and  that,  on 


18  MATHEMATICAL   PHILOSOPHY 

the  other  hand,  few  heads  are  naturally  so  clear  as  not 
to  require  discipline. 

Now,  it  so  happens  that  the  term  mathematics  is  the 
name  of  that  discipline  which,  because  it  attains  more 
nearly  than  any  other  to  the  level  of  logical  rigor,  is 
better  qualified  than  any  other  to  reveal  the  prototype 
of  what  is  best  in  the  quality  of  thinking  as  thinking. 
And  so,  in  accordance  with  the  principle  of  humanistic 
education,  we  have  to  say  that  the  amount  of  mathematical 
training  essential  to  the  appropriate  education  of  men  and 
women  as  human  beings  and  essential,  therefore,  to 
philosophers  as  human  beings,  is  the  amount  necessary 
to  give  them  a  fair  understanding  of  Rigor  as  the  standard 
of  logical  rectitude  and  therewith,  if  it  may  be,  the  spirit 
of  loyalty  to  the  ideal  of  excellence  in  the  quality  of 
thought  as  thought. 

Such  is  my  answer  to  the  question  that  has  detained 
us  so  long.  It  is,  you  observe,  a  qualitative  answer  in 
terms  of  a  great  ideal  and  a  sovereign  principle  of  edu- 
cation. If  I  must  add  a  word  touching  the  strictly  quan- 
titative aspect  of  the  question,  if  I  must,  that  is,  attempt 
to  indicate  the  extent  of  courses  and  the  length  of  time 
necessary  and  sufficient  to  yield  the  required  quality  and 
degree  of  training,  I  do  so  with  less  confidence  and  far 
less  interest.  For  so  much,  so  very  much,  depends  on  the 
pupil's  talent  and  the  quality  of  instruction.  A  consider- 
able degree  of  native  mathematical  ability  is  much  more 
common  than  is  commonly  supposed.  Born  mathematical 
imbeciles  are  rare.  Youth  of  fair  mathematical  talent 
constitute  an  immense  majority.  I  venture  to  say,  re- 
garding the  question  of  time  and  the  extent  of  courses, 
that,  for  pupils  of  fair  mathematical  endowment,  a  col- 
legiate freshman  year  or  even  a  high  school  senior  year 


INTRODUCTION  19 

of  geometry  and  algebra,  if  the  subjects  be  administered 
in  the  true  mathematical  spirit,  with  due  regard  to  pre- 
cision of  ideas  and  to  the  exquisite  beauty  of  perfect 
demonstration,  is  sufficient  to  give  a  fair  vision  of  the 
ideal  and  standard  of  sound  thinking. 

Herewith,  I  have  come  to  the  end  of  what  I  desired 
to  say  respecting  the  mathematical  equipment  essential 
to  a  philosopher  in  so  far  as  its  measure  depends  upon 
the  fact  that  philosophers  are  human  beings.  It  remains 
to  enquire  what  further  mathematical  attainments  are  to 
be  regarded  indispensable  to  the  proper  equipment  of  a 
philosopher  as  a  philosopher.  It  is  evident  that  the  an- 
swer must  be  sought  in  the  nature  of  the  philosopher's 
vocation.  It  would  be  presumptuous  in  me,  a  student 
of  mathematics,  to  offer  to  teach  you,  who  are  students 
of  philosophy,  the  nature  of  your  vocation,  but  I  may 
remind  you  of  it  for  it  is  necessary  to  have  it  clearly  in 
mind  if  we  are  to  see  its  bearings  upon  the  question  in 
hand.  No  one,  I  suppose,  has  conceived  the  philosopher's 
vocation  more  justly  and  nobly  or  characterized  it  more 
clearly  and  truly  than  Plato,  as  no  other  has  drawn,  with 
such  clarity  and  charm,  with  so  perfect  a  union  of  finesse 
and  amplitude,  so  beautifully  and  so  truly,  the  spiritual 
portrait  of  the  genuine  philosopher.  You  are,  of  course, 
familiar  with  the  characterization  and  the  portrait,  which 
together  give  for  all  time  a  vision  of  the  great  ideal: 
what  genuine  philosophy  is,  and  the  philosopher  ought 
to  be.  I  wish  to  remind  you  of  such  elements  of  it  as 
our  present  task  requires. 

The  genuine  philosopher,  says  Plato,  "has  magnifi- 
cence of  mind";  there  is  in  him  "no  secret  corner  of  illib- 
erality"  ;  he  is  "noble,  gracious,  the  friend  of  truth,  justice, 
courage,  temperance";  he  aims  at  being  "a  spectator  of 


20  MATHEMATICAL   PHILOSOPHY 

all  time  and  all  existence,"  and  so  he  is  a  lover  and  seeker 
of  "wisdom,"  which  does  not  consist  of  sense-impressions 
nor  of  "the  tempers  and  tastes  of  the  motley  multitude" 
nor  of  fickle  "blinking  opinion"  begotten  of  time-born 
appearances  and  events  destined  to  the  doom  of  thing9 
that  perish  in  "the  sea  of  change,"  but  consists  in  knowl- 
edge of  things  that  abide — of  true  being — of  whatsoever 
in  the  world  is  eternal:  pursuit  of  such  wisdom  is  the 
philosopher's  vocation,  sustained  by  the  twofold  hope 
of  coming  at  length  into  the  full-shining  presence  of  the 
Beautiful,  the  True,  and  the  Good  and  of  bringing  light 
from  them  into  the  lives  of  the  children  of  men. 

From  that  conception  of  the  genuine  philosopher's 
vocation  and  character,  what  conclusion  follows  regarding 
his  obligation  to  mathematics?  An  important  conclusion, 
as  I  hope  to  show  if  you  agree  with  me  in  thinking  that 
we  ought  to  ascertain  what  it  is. 

Let  me  say  at  the  outset  that  there  are  two  pretty 
obvious  considerations  which  I  do  not  intend  to  insist 
upon,  although  they  are  not  without  relevance  and  weight. 
One  of  them  is  that  which  conceives  mathematics  as  being 
itself  a  branch  of  philosophy;  the  other  relates  to  the 
familiar  contention  of  Plato,  that  mathematical  discipline 
is  indispensable  as  a  preparation  for  what  he  conceived 
to  be  the  philosopher's  distinctive  task — that  of  Dialectic. 

As  to  the  former  consideration,  one  might  argue, 
pertinently  and  confidently,  that  both  historically  and  in 
accordance  with  the  foregoing  conception  of  philosophy, 
Logic  is  one  of  its  branches;  that  mathematics  not  only 
employs  logic  as  an  instrument  but  is,  in  fact,  identical 
with  it,  mathematics  (as  traditionally  viewed)  being  re- 
lated to  logic  (as  traditionally  viewed)  as  the  trunk  and 
branches  of  a  tree  are  related  to  its  roots;  that,  conse- 


INTRODUCTION  21 

quently,  mathematics,  being  identical  with  logic,  is  not  ex- 
ternal to  philosophy  but  is,  strictly  speaking,  one  of  its 
principal  divisions;  and  that,  accordingly,  philosophers, 
if  they  are  not  to  be  ignorant  of  one  of  the  chief  depart- 
ments of  their  own  subject,  are  obliged  to  be,  not  merely 
mathematical  dilettanti,  but  mathematical  students,  seri- 
ous explorers  of  the  science.  Theoretically,  the  argument 
is  sound,  which  is  the  highest  quality  of  argument  as 
such.  I  do  not,  however,  as  I  have  said,  intend  to  press 
it,  because  it  imposes  on  the  student  of  philosophy  an 
obligation  that  he  cannot  fully  meet;  his  obligations  are 
many,  too  many  and  too  great;  he  may  not  reasonably 
hope  to  win  the  proper  competence  of  a  mathematician 
in  a  subject  where  the  developments,  still  rapidly  progress- 
ing in  numerous  directions,  have  already  reached  propor- 
tions so  great  that  no  man,  though  he  have  the  wide- 
reaching  arms  of  a  Henri  Poincare,  can  contrive  to  em- 
brace them  all. 

Turning  now  to  the  second  one  of  the  two  considera- 
tions mentioned  a  moment  ago,  let  me  guard  against  the 
danger  of  being  misunderstood.  You  are  aware  that,  in 
the  view  of  Plato,  what  is  peculiar  to  philosophy  is  dia- 
lectic— "the  coping  stone  of  the  sciences";  you  are  aware 
that  dialectic  is  the  sole  means  by  which  the  philosopher 
may  gain  a  knowledge  of  uwhat  each  thing"  in  the  hier- 
archy of  being  "essentially  is,"  and  by  which  he  may  gain, 
at  length,  as  he  ascends  the  scale,  a  vision  of  things  su- 
preme— absolute  justice,  absolute  beauty,  absolute  truth, 
absolute  good;  you  are  aware  that  the  successful  employ- 
ment of  dialectic  requires  not  only  native  "magnificence 
of  mind,"  but  also  a  long  course  of  preparation  in  the 
subsidiary  sciences;  you  are  aware  that,  according  to 
Plato,  the  most  indispensable  of  these  sciences  are  arith- 


22  MATHEMATICAL   PHILOSOPHY 

metic  and  geometry:  the  former  because  arithmetic,  not 
as  the  mere  practical  art  of  calculation  but  as  a  discipline 
in  the  logical  nature  of  pure  number,  "lays  hold  of  true 
being";  and  the  latter  because  "the  knowledge  at  which 
geometry  aims  is  knowledge  of  the  eternal."  Such  is  in 
brief,  as  you  know,  the  famous  contention  of  Plato  re- 
specting the  importance  of  mathematical  discipline  as  a 
preparation  for  philosophy.  There  can  be  no  doubt  that 
the  contention  is  perfectly  just.  Why,  then,  do  I  not 
stress  it  in  this  connection?  The  reason  is  that  the  mathe- 
matical discipline  insisted  on  by  Plato  is  more  than  cov- 
ered by  the  mathematical  training  I  have  already  urged  as 
essential  to  the  appropriate  education  of  the  philosopher 
as  a  human  being,  and  that  we  are  here  considering  such 
further  mathematical  attainments  as  are  essential  to  him 
as  a  philosopher.  Before  leaving  this  theme,  however, 
I  desire  to  point  out  a  different  aspect  of  it  and  in  connec- 
tion therewith  to  speak  very  briefly,  in  passing,  of  a  mat- 
ter which  I  have  discussed  elsewhere,1  to  which  I  hope  to 
return  at  a  later  stage  of  these  lectures  and  which,  I 
believe,  has  a  very  important  bearing  upon  the  question 
before  us. 

After  some  years  of  reflection,  I  am  convinced  that 
the  great  Platonic  Absolutes,  whose  "perception  by  pure 
intelligence"  brings  us,  says  Plato,  to  "the  end  of  the  in- 
tellectual world" — have  indeed  their  proper  locus  beyond 
it.  I  am  convinced  that,  instead  of  being  genuine  con- 
cepts amenable  as  such  to  the  logical  processes  valid  in 
the  intellect's  world,  the  Platonic  Absolutes  are  radiant 
ideals  of  concepts,  shining  from  above  them  like  down- 
ward-looking   aspects    of    an    over-world;    transcending 

1  Science  and  Religion,  also  The  New  Infinite  and  the  Old  Theology. 
Yale   University  Press. 


INTRODUCTION  23 

every  type  of  excellence  in  which  intellectual  progress  is 
possible,  they  appear  as  ideals  supernal — as  stars  beyond 
the  sky.  I  need  not  say  that  the  Absolutes,  thus  regarded, 
retain  their  glory  unimpaired  and  their  previous  value 
as  sources  of  light  and  inspiration.  We  should  not,  how- 
ever, fail  to  see  clearly  that,  if  they  be  thus  regarded,  the 
philosopher  is  thereby  confronted  by  a  new  challenge,  a 
new  problem,  a  new  field  of  study  or,  perhaps  I  should 
say,  by  an  old  one  seen  as  new.  For,  if  the  Absolutes 
are  not  in  the  intellectual  world  but  are  beyond  it;  if  they 
be,  in  fact,  not  concepts,  but  ideals  of  concepts,  shining 
downward  from  above  them,  then  obviously  their  origin, 
the  manner  and  genesis  of  their  appearance,  and  their 
significance  for  life,  must  be  sought  in  the  nature  and 
function  of  that  strange  and  familiar  spiritual  process 
omnipresent  among  the  activities  of  the  intellectual  world 
and  known  as  Idealization.  And  now  the  point  I  am 
aiming  at  and  to  which  I  invite  your  special  attention  is 
this:  In  the  study  of  this  great  subject — the  nature  and 
function  of  Idealization — the  philosopher  and  especially 
the  theologian  as  philosopher — for  rational  theology, 
rightly  conceived,  is  the  science  of  Idealization — will 
have  need  of  mathematical  discipline  surpassing  the 
Platonic  requirement  and  surpassing  what  I  have  deemed 
essential  to  the  education  of  the  philosopher  as  a  human 
being.  For  the  term  "idealization"  is  the  generic  literary 
term  for  what  in  science  and  especially  in  mathematics  is 
known  as  generalization  by  means  of  the  method  or 
process  of  limits.  In  mathematics,  particularly  in  the 
modern  theory  of  the  Real  Variable,  in  connection  with 
the  generalization  of  the  number  concept,  the  essential 
nature  of  Idealization,  the  pattern  of  it  as  the  process 
and  method  of  directing  the  attention  from  within  a  given 


24  MATHEMATICAL   PHILOSOPHY 

domain  of  operation  to  the  existence  and  the  character 
of  outlying  domains,  comes  into  perfect  light.  It  is  in 
mathematics  and  not  elsewhere  that  Idealization  is  be- 
held in  its  purity;  and  unless  the  philosopher  becomes 
familiar  with  it  there  in  its  purity,  his  endeavor  to  study 
the  great  process  elsewhere,  amid  the  many  disguises  half 
concealing  its  subtle  ramifications  throughout  the  shadowy 
world  of  general  thought,  will  encounter  serious  difficul- 
ties, if  not  defeat. 

The  considerations  I  have  now  advanced,  though  they 
are  subordinate,  are  weighty,  and  I  commend  them  as 
worthy  of  your  further  reflection.  Let  us  proceed,  with- 
out further  delay,  to  the  heart  of  the  matter. 

We  have  seen  that  the  genuine  philosopher  "has  mag- 
nificence of  mind";  that  there  is  in  him  "no  secret  corner 
of  illiberality" ;  that  his  vocation  requires  him  to  be  "a 
spectator  of  all  time  and  all  existence" ;  and  that  the  wis- 
dom he  seeks  is  the  wisdom  which  consists  in  knowledge 
of  whatsoever  is  eternal.  It  is  these  great  things — the 
highest  distinctive  marks  of  the  genuine  philosopher — 
that  determine  the  character  of  his  mathematical  obliga- 
tions and  enable  us  to  measure  them.  For  what  is  mathe- 
matics ?  What  is  that  science  which  Plato x  called  "divine," 
which  Goethe  called  "an  organ  of  the  inner  higher  sense," 
which  Novalis  called  "the  life  of  the  gods,"  and  which 
Sylvester  called  "the  Music  of  Reason"?  The  question 
is  not  intended  to  call  for  a  complete  description  of  the 
science,  much  less  for  a  definition  of  it.  What  it  seeks  is 
a  partial  description.  I  wish  merely  to  draw  your  atten- 
tion to  one  feature  of  mathematics — to  that  feature  of 
it  which  all  competent  judges  agree  in  signalizing  as  the 
chief  aspect  of  the  science  viewed  as  an  enterprise.  The 
aspect  in  question  I  endeavored  to  point  out  some  years 

1  See  Memorabilia  Mathematica  by  Professor  Moritz. 


INTRODUCTION  25 

ago  in  the  following  words:  "As  an  enterprise,  mathe- 
matics  is  characterized  by  its  aim,  and  its  aim  is  to  think 
rigorously  whatever  is  rigorously  thinkable  or  whatever 
may  become  rigorously  thinkable  in  course  of  the  upward 
striving  and  refining  evolution  of  ideas."  *  The  same 
feature  has  been  recently  indicated,  even  more  clearly 
perhaps  and  somewhat  poignantly,  in  a  striking  utterance 
by  Mr.  Bertrand  Russell.  "Pure  logic,  and  pure  mathe- 
matics (which  is  the  same  thing),  aims  at  being  true,  in 
Leibnizian  phraseology,  in  all  possible  worlds  and  not 
merely  in  this  higgledy-piggledy  job-lot  of  a  world  in 
which  chance  has  imprisoned  us."  2 

You  know,  at  least  in  a  general  way,  that  in  pursuit 
of  that  enterprise  and  aim  through  the  centuries,  the 
mathematical  spirit  has  achieved  immense  results  and  that 
today  the  science  of  mathematics,  as  a  body  of  permanent 
knowledge  regarding  things  eternal,  is  a  veritable  conti- 
nent of  expanding  doctrine.  And  so  it  is  pertinent  to  ask: 
How  can  one  aspiring  to  be  a  philosopher,  unless  he  ex- 
plores that  growing  continent  of  knowledge  respecting 
what  is  "true  of  all  possible  worlds,"  be  in  any  proper 
sense  "a  spectator  of  all  time  and  all  existence"?  You 
may  wish  to  reply  that,  owing  to  his  other  obligations,  the 
philosopher  cannot  make  the  exploration  fully;  that  in- 
deed, owing  to  the  nature  of  the  continent,  he  cannot, 
without  exploring  it  step  by  step,  gain  even  so  much  as  a 
clear  knowledge  of  its  contour  and  relief;  that,  however, 
notwithstanding  the  endless  diversity  of  the  things  that 
are  there,  they  have  a  certain  essential  character  in  com- 
mon; that  for  the  philosopher's  vocation,  knowledge  of 

1  Human    Worth    of   Rigorous    Thinking,   p.    3.      Columbia    University 
Press. 

2  Introduction  to  Mathematical  Philosophy.    The  Macmillan  Company, 
New  York. 


26  MATHEMATICAL    PHILOSOPHY 

that  common  character  is  sufficient;  and  that  such  knowl- 
edge does  not  demand  exploration  of  the  continent  in  all 
its  length  and  breadth  and  height  and  depth,  but  may  be 
gained  by  examination  of  representative  parts  and  especi- 
ally of  the  elements  which  fundamentally  compose  the 
whole . 

That  reply,  if  we  rightly  interpret  the  meaning  of 
the  terms,  is  just.  But  their  meaning  is  momentous.  The 
mathematical  knowledge  which  they  tell  us  is  sufficient 
for  the  purposes  of  the  philosopher  is  neither  slight  nor 
simple  nor  easy  to  gain.  The  questions  it  must  answer 
determine  its  nature  and  its  scope.  What  are  the  idiosyn- 
crasies of  mathematics  as  a  body  of  content?  As  a  sys- 
tem of  methods?  As  a  type  of  activity?  As  a  distinct- 
ive enterprise  among  the  great  kindred  enterprises  of  the 
human  spirit?  If  the  science  be  logical,  what  are  its  re- 
lations to  Logic?  If  it  be  beautiful,  what  are  its  rela-( 
tions  to  Art?  If  it  employ  hypothesis,  observation  and 
experiment,  what  are  its  relations  to  Natural  Science?  If 
it  be  purely  abstract  and  conceptual,  what  are  its  relations 
to  the  concrete  world  of  Sense?  If  it  be  theoretic,  what 
are  its  relations  to  Practical  Life?  If  it  be  self-critical, 
what  are  its  relations  to  the  science  and  art  of  Criticism? 
If  it  be  a  wisdom  respecting  infinite  and  eternal  things, 
what  are  its  relations  to  Philosophy  and  to  Religion?  If 
it  have  limitations,  what  are  its  relations  to  the  dream 
of  Universal  Knowledge?  To  the  challenge  of  these 
great  questions  and  their  kind,  no  one  having  "magnifi- 
cence of  mind,"  no  one  called  to  be  "a  spectator  of  all 
time  and  all  existence,"  can  fail  to  respond.  And  so  we 
see  that  the  mathematical  obligations  of  the  philosopher 
confront  him  with  two  difficult  close-related  Problems: 
the  problem  of  definition  and  the  problem  of  evaluation; 


INTRODUCTION  27 

he  must  endeavor  to  ascertain  what  mathematics  essen- 
tially is  and  endeavor  to  estimate,  in  the  terms  of  spiritual 
Worth,  the  rank  and  the  dignity  of  the  science  in  the 
hierarchy  of  knowledges  and  arts. 

It  is  a  radical  error  to  regard  these  kindred  tasks  of 
definition  and  evaluation  as  belonging  to  the  proper  func- 
tion of  mathematicians  as  such.  The  term  "mathematics" 
is  the  name  of  an  immense  class  of  logically  related  terms 
and  most  of  these  the  mathematician  must  indeed  de- 
fine, but  the  term  "mathematics,"  which  names  the  class, 
is  not  among  them;  the  class  is  not  a  member  of  itself, 
for  no  class  can  be;  the  name  "mathematics"  is  not  a 
mathematical  term;  the  mathematician  would  be  none  the 
less  a  mathematician,  had  he  never  heard  of  it;  it  is  a 
philosophical  term,  used  by  mathematicians  as  a  conven- 
ience but  never  as  a  necessity.  The  proper  activity,  the 
distinctive  function,  of  the  mathematician  is  to  mathemat- 
icize,  as  that  of  a  swimmer  is  to  swim;  or  that  of  a 
farmer,  to  farm;  or  that  of  a  poet,  to  make  poetry;  or 
that  of  a  trader,  to  trade.  And  it  is  as  little  the  business 
of  the  mathematician  to  define  and  evaluate  the  peculiar 
type  of  his  proper  activity  as  it  is  that  of  the  swimmer 
or  the  farmer  or  the  poet  or  the  trader  to  do  the  like  for 
his.  The  philosopher,  therefore,  may  not  rightly  look 
to  mathematicians  as  such  for  a  definition  of  mathematics 
nor  for  any  appraisement  of  its  significance  or  its  worth. 

Is  it  not  true,  nevertheless, — you  may  wish  to  ask — 
that  nearly  all  real  advancement  made  in  the  course  of 
the  centuries  in  these  tasks  of  definition  and  appraise- 
ment has  been  made  by  mathematicians?  The  answer  is 
yes,  even  if  we  do  not  forget  or  underrate  the  relevant 
contributions  of  Plato  and  Aristotle,  for  knowing,  as  they 
did,  what  was  known  then  of  mathematics,  they  must  be 


28  MATHEMATICAL   PHILOSOPHY 

counted  among  the  mathematical  scholars  of  their  day. 
It  must  be  noted,  however,  that,  though  the  advancement 
in  question  was  made  by  mathematicians,  it  was  made  by 
them,  not  in  their  character  as  mathematicians,  but  in 
their  capacity  as  philosophers.  There  is  nothing  in  the 
fact  to  astonish.  For  a  man  is  greater  than  any  occupa- 
tion, and  a  mathematician,  like  a  physician  or  lawyer  or 
poet  or  statesman  or  farmer,  may  be — indeed  he  must  be, 
in  some  measure — a  philosopher  as  well.  It  is  not,  then, 
strange  or  a  matter  for  wonder  that  there  have  been 
mathematicians  who,  in  relation  to  their  proper  subject 
taken  as  a  distinctive  whole,  have  sometimes  taken  the 
attitude  and  played  the  role  of  philosopher.  Nay,  even 
within  the  subject,  in  relation  to  its  parts,  the  role  is  very 
common;  for  whenever  a  mathematician,  having  acquired 
competence  in  two  or  more  branches — say  algebra  and 
geometry — pauses  to  compare  them,  seeking  to  ascertain 
the  essential  nature  of  each,  what  they  have  in  common, 
their  respective  worths  and  their  joint  significance  as 
forms  of  activity,  his  interest  and  his  attitude  have  then 
become  for  the  time,  whether  long  or  short,  those  of  the 
philosopher.  The  fact. is  that  such  minor  alternations  of 
the  scientific  and  the  philosophic  interests  may  be  con- 
stantly witnessed  even  in  the  activity  of  such  mathema- 
ticians as  ignorantly  affect  to  spurn  philosophy  and  to 
scorn  its  achievements;  but  they  are  not  aware  of  it. 

Of  the  two  tasks  with  which,  as  we  have  seen,  the 
mathematical  obligations  of  the  philosopher  confront  him, 
the  task  of  definition  is  far  more  advanced  than  that  of 
evaluation;  and,  though  the  work  of  the  former  is  not  yet 
complete,  we  know  much  better  today  what  mathematics 
is  than  what  it  is  worth.  That  it  should  be  so  is  natural, 
for  a  just  appraisement  of  worth  depends,  of  course,  upon 


INTRODUCTION  29 

the  nature  of  the  thing  appraised.  We  are,  therefore, 
not  surprised  to  find  that  researches  concerning  the  essen- 
tial nature  of  mathematics  have  been  prosecuted,  espe- 
cially in  recent  times,  far  more  resolutely  and  systemat- 
ically than  such  as  aim  at  a  critical  estimate  of  its  sig- 
nificance and  value.  In  Plato  and  in  Aristotle,  as  you 
know,  research  of  both  kinds  produced  results  of  great 
importance.  I  shall  not  speak  of  the  great  Greek  mathe- 
maticians for  their  interest  centered,  not  in  the  philosophy 
of  their  subject,  but  in  the  science  of  it.  They  were  swim- 
mers mainly — not  non-aquatic  students  of  swimming.  It 
seems  incredible  that,  after  Plato  and  Aristotle,  no  im- 
portant contribution  to  the  philosophy  of  mathematics  was 
made  in  the  course  of  twenty  hundred  years.  Yet  that 
is  the  fact.  Even  the  brilliant  and  exquisite  De  L'Esprit 
Geomelrique  of  Pascal  is  thoroughly  Aristotelian.  The 
great  revival  had  to  await  the  appearance  of  Leibniz 
— of  him  who  said,  "Ma  metaphysique  est  toute  mathe- 
matique."  As  students  of  philosophy,  you  know  that 
throughout  his  life  this  marvelous  man  was  haunted  by  a 
magnificent  dream — the  dream  of  "a  universal  mathe- 
matics." In  his  manifold  endeavors  to  make  the  dream 
come  true  is  found  the  origin  of  that  great  critico-con- 
structive  movement  which  has  done  more  than  all  previ- 
ous centuries  to  disclose  the  essential  nature  of  rigorous 
thought  and  which,  after  notable  vicissitudes  of  fortune, 
is  known  today,  in  all  scientific  countries  of  the  world, 
under  the  characteristic  name  of  Symbolic  Logic. 

The  leading  names  of  its  pioneers  and  contributors — 
Leibniz,  Lambert,  De  Morgan,  Boole,  Jevons,  Schroder, 
Peirce  (C.  S.),  MacCall,  Frege,  Peano,  Russell,  White- 
head, Hilbert,  Huntington,  Couturat,  and  others — suf- 
ficiently indicate  its  international  interest  and  the  variety 


SO  MATHEMATICAL    PHILOSOPHY 

of  genius  to  which  it  appeals.  The  growing  literature 
of  the  subject  is  large.  Fortunately,  it  is  not  necessary, 
except  for  the  historian,  to  examine  it  all,  for  it  has  been 
refined,  assimilated,  and,  all  but  the  later  developments, 
superseded  in  the  monumental  work  of  Whitehead  and 
Russell — Principia  Mathematica — the  present  culmina- 
tion of  the  movement.  This  work,  however,  which  ha9 
not  yet  been  completed,  the  philosopher  must  examine 
minutely  if  he  would  understand,  as  a  philosopher  ought 
to  understand,  the  fundamental  nature  of  mathematics  as 
disclosed  in  the  best  light  that  has  been  thrown  upon  it 
and  especially  if  he  would  realize  the  hope  of  being  able 
to  improve  the  light,  which  is  not  yet  perfect.  The 
symbols  are  at  first  repellent;  they  tend  to  frighten  but 
are  not  in  fact  difficult  to  master. 

They  are  things  of  so  frightful  mien 
That  to  be  hated  need  only  be  seen. 
But  often  seen,  familiar  with  their  face, 
We  endure  them  first  and  then  embrace. 

Theoretically,  the  symbols  are  not  essential,  a  suf- 
ficiently powerful  god  could  get  along  without  them;  but 
practically  they  are  indispensable  as  instruments  for 
economizing  our  intellectual  energy.1 

No  kind  of  work,  whether  philosophic  or  scientific, 
can  be  severer  in  its  demands.  None  surpasses  it  in  re- 
spect of  the  toil  involved,  nor  in  patience,  nor  in  depth 
of  penetration,  nor  in  subtlety,  nor  imagination,  nor  an- 
alytic finesse,  nor  in  the  demand  it  makes  upon  the  con- 
structive faculty,  and  none  can  give  to  the  competent 
student  a   serener  vision  of  eternal  things.     If  on  this 

1  In  relation  to  the  early  history  and  importance  of  symbolism  do  not 
fail  to  read  Professor  David  Eugene  Smith's  beautiful  essay,  "Ten  Great 
Epochs  in  the  History  of  Mathematics,"  in  Scientia,  June,  1931. 


INTRODUCTION  31 

account  it  seems  to  you,  as  It  may  seem,  a  little  strange 
that  the  majority  of  mathematicians  have  little  interest 
in  such  work  and  are  not  familiar  with  it,  it  is  sufficient 
to  reflect  that,  though  its  results  as  results  are  strictly 
scientific,  strictly  a  part  of  mathematics,  they  are  deeply 
tinged  with  philosophic  interest  and  owe  their  discovery 
primarily  to  the  spirit  of  philosophic  enquiry.  In  mathe- 
matics, as  in  other  subjects,  fashions  change;  it  is,  more- 
over, so  large  a  subject  that  a  student  is  obliged  by  his 
limitations  to  specialize  in  a  branch  of  it  or  in  a  group 
of  branches;  and  it  so  happens  that  a  large  majority  of 
mathematicians  are  disqualified, — some  of  them  by  breed- 
ing, more  of  them  by  temperament, — for  study  or  re- 
search in  that  branch  which  deals  with  the  foundations  of 
their  science  as  a  whole.  Such  disqualification  is  not  to 
be  imputed  to  them  as  a  fault;  often  no  doubt, — oftener 
than  not,  perhaps, — it  is  only  a  defect  of  a  quality;  at  all 
events,  a  mathematician  may  not  be  rightly  blamed  for 
the  temperamental  bent  of  his  scientific  interests.  The 
same  may  not  be  said  of  those  who  are  inclined  to  de- 
preciate other  interests  than  their  own.  I  refer  to  the 
type  of  mathematician, — such  as  you  may  sometimes 
meet, — who,  as  if  to  mitigate  his  sense  of  guilt  for  being 
consciously  innocent  of  symbolic  logic  and  so  to  protect 
his  self-respect,  will  occasionally  ask  you,  in  a  somewhat 
disparaging  tone,  to  tell  him,  if  possible,  of  any  important 
service  rendered  by  symbolic  logic  or  of  any  important 
proposition  established  by  it  or  of  any  important  method 
devised  by  it  for  the  use  of  mathematicians.  If  you  dis- 
regard the  spirit  in  which  such  questions  are  sometimes 
asked,  it  is  easy  to  answer  them  in  a  way  satisfactory  to 
any  candid  and  competent  enquirer.  7hc  answer,  as  I 
conceive  it,  is,  in  brief,  as  follows: 


32  MATHEMATICAL   PHILOSOPHY 

(i)  Symbolic  logic  has  established  the  thesis  that 
all  existing  mathematics  (and  presumably  all  potential 
mathematics)  is  literally  a  logical  outgrowth  of  a  few 
primitive  ideas,  and  a  few  primitive  propositions,  of 
logic;  and,  that,  accordingly,  logic  and  mathematics 
are  spiritually  one  in  the  sense  in  which  the  roots,  the 
trunk  and  the  branches  of  a  tree  are  physically  one:  a 
proposition  which,  though  philosophical  and  not  mathe- 
matical, is,  in  respect  of  human  significance,  unsur- 
passed. 

(2)  In  course  of  the  work  establishing  the  fore- 
going proposition,  symbolic  logic  has  discovered  and 
rigorously  demonstrated  a  long  sequence  of  theorems 
respecting  propositions,  classes,  and  relations,  which 
theorems  constitute  an  immense  new  body  of  genuinely 
mathematical  doctrine  underlying  mathematics  as  com- 
monly understood  and  they  are  open  to  inspection  by 
all  critics,  whether  friendly  or  unsympathetic. 

(3)  Symbolic  logic  has  not  promised  nor  pretended 
to  devise  methods  to  facilitate  mathematical  research 
except  research  in  mathematical  foundations;  in  such 
research  the  effectiveness  of  the  methods  employed  is 
patent  in  the  results. 

(4)  Finally,  symbolic  logic  is  simply  the  latest 
fruit  of  the  critical  spirit  in  mathematics — fruit  of  the 
refinement, — the  inevitable  refinement, — of  that  spirit 
which  has  led  to  so  many  mathematical  developments 
familiar  to  all  mathematicians, — the  postulational 
method,  for  example,  the  birth  of  non-Euclidean  geo- 
metries, the  theory  of  manifolds  including  the  hyper- 
spaces,  the  so-called  arithmetization  of  mathematics, 
and  similar  phenomena  throughout  the  history  of  the 
science.     To  depreciate  symbolic  logic  is  to  oppose  the 


INTRODUCTION  S3 

progress  of  the  spirit  of  constructive  criticism  and  that 
means  opposition  to  the  progress  of  science;  for 
Cousin's  famous  mot  is  just :  La  critique  est  la  vie  de  la 
science. 

In  saying  that  the  philosopher's  mathematical  obliga- 
tions require  him  to  familiarize  himself  with  the  methods 
and  results  of  symbolic  logic,  I  have  not  quite  finished  the 
tale.  One  point  remains  to  be  stressed.  Before  present- 
ing it,  let  me  remind  you  of  a  certain  fairly  obvious  dis- 
tinction which  Bergson  1  has  emphasized  and  has  elevated, 
rightly  I  believe,  to  the  level  of  an  important  principle 
of  knowledge.  I  may  best  make  it  clear  by  an  example. 
You  know,  as  we  say,  how  to  move  your  arms.  This 
knowledge  is  not  a  part  of,  and  is  not  derived  from, 
your  "scientific"  knowledge  of  physiology,  anatomy  and 
physics,  though  this  knowledge,  too,  may  tell  you  much 
respecting  the  motion  in  question.  The  latter  knowledge 
is  indirect  and  external — a  knowledge  from  without;  the 
former  is  immediate  and  internal — a  knowledge  from 
within;  it  is  a  living  instinct — of  the  essence  of  your  life; 
the  other  is  only  a  superadded  understanding.  Complete 
or  perfect  knowledge  of  any  thing  involves  both  of  these 
kinds  of  knowledge.  In  the  illustration  I  have  used,  the 
thing  to  be  known  is  a  part  of  the  knower — the  mobile 
arm  is  yours  and  its  life  is  yours.  But  most  objects  of 
knowledge  are  not  thus  parts  of  the  knower.  Of  such 
objects  complete  knowledge,  even  if  we  suppose  the  ele- 
ment of  "understanding"  to  be  perfectible,  is  unattain- 
able; for  to  attain  it,  to  gain  the  other  element, — the  in- 
stinctive  element,    the  inner  kind   of  knowlege, — would 

1  "Introduction    a    la    M^taphysique."     Revue   de   Melaphysique   el  de 
Morale,  Vol.  n,   (1903). 


34  MATHEMATICAL   PHILOSOPHY 

require  the  knower  to  make  the  object's  life  an  intimate 
part  of  his  own;  and  this,  it  is  plain,  cannot  be  done  per- 
fectly. But — and  here  is  a  fact  of  the  utmost  importance 
— it  can  be  done  approximately.  Do  you  ask,  how?  The 
answer  is:  By  the  noetic  agency  of  sympathy  or  love; 
by  the  means  which  Bergson  has  so  finely  described  as 
"intellectual  sympathy"  with  the  object's  life.  Your 
thought,  I  fancy,  runs  ahead  of  my  speech  and  already 
sees  the  bearing  of  the  point  upon  the  philosopher's  ob- 
ligations to  mathematics.  In  a  sense  more  than  figurative, 
this  science  has  a  life  of  its  own.  Else  how  could  it  grow? 
To  acquire  such  knowledge  of  the  science  as  the  philoso- 
pher's vocation  demands,  to  know  it  from  within  as  it 
instinctively  knows  itself,  he  must  acquire  such  intellectual 
sympathy  with  it  as  will  enable  him  to  feel  its  proper  life 
as  part  of  his  own.  Sympathy  so  living  and  intimate, — 
embracing  the  instincts,  and  feeling  the  impulses  and 
moods,  of  an  alien  life, — is  not  easily  acquired.  In  the 
case  of  mathematics,  collegiate  courses  in  algebra, 
geometry  and  trigonometry  cannot  give  it,  except  to  the 
born  mathematician,  who  has  it  already;  neither  can  it 
be  given  adequately  by  symbolic  logic  for  this  study  is  too 
meditative  for  the  purpose,  too  introspective,  being  more 
concerned  to  "understand,"  than  to  "live,"  the  life  of 
mathematics.  No,  if  the  student  of  philosophy  would 
acquire  that  kind  of  knowledge  of  mathematics  which  can 
come  to  him  only  through  intellectual  sympathy  with  its 
life,  he  must  share  its  life;  he  must  penetrate  it  deeply 
enough  to  feel  the  touch  and  thrill,  the  push  and  sweep, 
of  its  conquering  tide;  he  must  at  least  plunge  into  Analy- 
tical Geometry  and  the  Infinitesimal  Calculus  where  the 
science  first  won,  and  its  votaries  first  win,  a  worthy  sense 
of  its  power  and  its  destiny. 


INTRODUCTION  35 

In  the  light  of  the  foregoing  considerations,  the 
mathematical  obligations  of  the  philosopher  appear  to  be 
heavy.  They  are  heavy;  but  they  are  not  too  heavy  for 
those  whose  native  talents  qualify  them  for  a  vocation 
demanding  "magnificence  of  mind."  It  is  consoling  to 
know  that  a  student  who  faithfully  keeps  the  obligations 
will  have  two  great  rewards:  the  joy  of  an  insight  and  a 
power  not  to  be  otherwise  gained;  and  the  joy  of  repre- 
senting and  perpetuating  a  noble  tradition  of  his  kind, — 
the  tradition,  I  mean,  of  mathematical  competence  as  il- 
lustrated by  the  heroes  of  philosophy  in  every  important 
age.  In  relation  to  that  tradition,  it  is  indeed  true,  as  you 
know,  that  there  have  been  many  philosophers  of  great 
learning,  some  of  them  important  thinkers,  whose  ignor- 
ance of  mathematics  has  been  virtually  complete,  and 
these  have  differed  widely  in  kind;  of  their  mathematical 
ignorance  some  of  them  have  not  been  aware;  some  have 
deeply  regretted  it  and  humbly  confessed  it — our  own  be- 
loved William  James,  for  example;  in  some  it  has  been 
not  only  complete  but  shameless  as  well,  even  haughty 
and  defiant,  as  in  Sir  William  Hamilton  and  Schopen- 
hauer, whose  false  and  malicious  diatribes  against  mathe- 
matics I  have  dealt  with  elsewhere,1  and  in  case  also,  I 
am  sorry  to  say,  of  Benedetto  Croce,"  whose  fine  literary 
and  artistic  culture  and  true  elevation  of  spirit  have  not 
availed  to  restrain  him  from  speaking  with  strange  con- 
fidence and  very  disparagingly  of  a  science  which  his  fel- 
low countrymen,  by  brilliant  research,  have  done  so  much 
to  honor  and  which  he  has  not  qualified  himself  to  under- 
stand even  slightly. 

It  is  edifying  to  compare  such  representatives  of  phi- 

1  Human  Worth  of  Rigorous  Thinking,  p.  290. 
1  Logic  as  the  Science  of  the  Pure  Concept. 


S6  MATHEMATICAL   PHILOSOPHY 

losophy  with  its  towering  heroes,  its  men  of  "summit- 
minds"  :  with  Plato,  for  example,  who  knew  perfectly  the 
mathematics  of  his  time,  whose  sense  and  revelation  of 
its  spiritual  significance  has  never  been  surpassed,  and 
whose  influence  in  his  own  and  all  succeeding  ages  has 
given  his  name  a  permanent  place  in  mathematical  history; 
and  with  Aristotle,  whose  discussions  of  such  fundamental 
questions  as  the  nature  of  mathematical  definition,  hy- 
pothesis, axiom,  postulate,  and  subject  matter,  are  of  high 
value  even  today  and  whose  great  contributions  to  logic 
must  now  be  regarded,  in  the  light  of  modern  symbolic 
logic,  as  being,  though  he  did  not  know  it,  genuine  con- 
tributions to  mathematics;  and  with  Descartes,  discoverer 
of  important  mathematical  propositions,  and  chief  inven- 
tor of  analytical  geometry, — second  in  scientific  power  to 
only  one  among  mathematical  methods;  and  with  Leibniz, 
father  of  modern  symbolic  logic  and  co-inventor  with 
Newton  of  the  infinitesimal  calculus,  "the  most  powerful 
instrument  of  thought  yet  devised  by  the  wit  of  man"  ;a 
and  with  Spinoza  to  whose  lot  it  fell  to  try  the  great  ex- 
periment,— inevitable  in  the  history  of  thought, — of  cloth- 
ing ethical  theory, — highest  of  human  interests, — with 
the  strength  and  beauty  of  mathematical  rigor  and  form, 
and,  in  trying  it,  to  exemplify  in  a  singularly  noble  way, 
the  fact  that  illustrious  failures  fall  to  the  lot  of  none 
but  illustrious  men;  and  with  other  great  philosophic 
personalities,  if  I  did  not  fear  to  weary  you  in  naming 
them,  who  by  their  mathematical  competence  worthily 
represent  the  heroic  tradition. 

In  closing  this  initial  lecture,  I  desire  to  indicate  in  a 
general  way  the  sort  of  topics  with  which  the  following 
lectures  will  deal.     The  endless  number  of  the  ideas,  or 

1  See  the  preface  of  Professor  W.  B.  Smith's  Infinitesimal  Calculus. 


INTRODUCTION  37 

notions,  or  concepts, — as  they  are  variously  called, — 
which  enter  as  components  into  the  stately  edifice  of 
mathematics,  though  they  are  all  of  them,  in  a  sense,  in- 
dispensable to  it,  yet  differ  very  widely  in  respect  of  their 
place  and  rank,  their  dignity  and  structural  service.  Ex- 
amination of  the  great  edifice  makes  it  evident  that  some 
of  them, — a  relatively  small  number  of  them, — have  the 
distinction  of  being  related  to  it  as  central  supporting 
pillars.  Among  the  chief  of  these  are  the  concepts  de- 
noted by  the  terms:  Function — Propositional  Function 
— Implication — Proposition — Class — Relation — Postulate 
System — Doctrinal  Function — Doctrine — Variable — Limit 
— Number — Finitude — Infinity — Transformation — Group 
— Invariance.  It  is  with  such  pillar-concepts, — which 
are  obviously  not  coordinate  in  rank, — that  I  purpose 
to  deal,  and  I  shall  deal  with  them  primarily  as  concepts, 
explaining  them  with  constant  regard  to  clarity,  with  a 
minimum  of  technical  symbols,  and  with  a  view,  not 
alone  to  their  mathematical  meanings,  but  to  their  sig- 
nificance and  use  in  outlying  fields  of  thought.  But  I  shall 
not  endeavor  to  expound,  in  the  proper  sense  of  the 
term,  the  great  technical  doctrines  that  have  grown  up 
about  them  as  subject  matter,  for  such  exposition  would 
demand,  as  you  know,  not  merely  one  course,  but  many 
courses,  of  lectures.  You  will  rightly  infer  that,  though 
proof  or  demonstration  may  not  be  entirely  absent,  it  will 
not  be  permitted  to  detain  us  too  long,  much  less  to 
dominate  the  discussions. 

Let  me  say,  finally,  that  the  course  is  not  designed  to 
be,  in  the  stricter  and  narrower  sense  of  the  term,  a  course 
in  the  philosophy  of  mathematics.  It  aims  at  being  at 
once  something  less  and  something  more:  less,  in  that  it 
does  not  endeavor  to  begin  with  the  most  ultimate  of 


38  MATHEMATICAL   PHILOSOPHY 

logical  principles  and  to  build  upon  them,  little  step  by 
step,  with  infinite  patience,  the  solid  masonry  of  the  mathe- 
matical edifice;  more,  in  that  it  is  a  good  deal  concerned 
with  the  mentioned  task  of  evaluation — with  disclosing 
the  relations  of  mathematics  to  other  great  forms  of  in- 
tellectual activity  and  especially  its  bearings  upon  the  uni- 
versal interests  of  the  human  spirit. 


LECTURE  II 
Postulates 

CONCRETE     DEFINITION     OF     POSTULATE     SYSTEM THE 

PROTOTYPE    OF     PRINCIPLES     OR     PLATFORMS THE 

ANCIENT    "CRAFT    OF    GEMETRY" THE    SWORD    OF, 

THE     GADFLY CLARITY     OR     SILENCE MUNICIPAL 

LAWS  AND  THE  LAWS  OF  THOUGHT. 

The  introductory  lecture  has  served,  I  hope,  to  indi- 
cate in  a  general  way  the  aim,  the  spirit,  and  the  scope  of 
our  undertaking.  In  deciding  to  begin  the  work  proper  with 
a  study  of  the  great  concept  denoted  by  the  familiar  term 
— Postulate  System — I  have  been  guided  by  three  con- 
siderations: ( I )  every  question  arising  in  what  is  strictly 
called  the  philosophy  of  mathematics — in  the  study,  that 
is,  of  its  logical  foundations — is  connected  more  or  less 
closely,  directly  or  indirectly,  with  that  concept,  which  is 
thus  the  central  ganglion  of  mathematical  philosophy; 
(2)  by  means  of  the  concept  in  question  and  without  un- 
necessary delay,  I  desire  to  set  in  clear  light  another  con- 
cept, intimately  related  to  it,  to  which  I  have  given  the 
name — Doctrinal  Function — and  which,  if  I  am  not  mis- 
taken, has  great  philosophic  importance;  (3)  postulate 
systems  as  employed  in  mathematics,  appear  there  in 
perfect  light  as  systems  of  principles  underlying  and  sup- 
porting definite  bodies  of  thought,  and  so  they  serve  as 
a  model,  as  an  ideal  prototype,  for  the  inspiration,  the 

39 


40  MATHEMATICAL    PHILOSOPHY 

guidance  and  the  criticism  of  every  rational  enterprise, 
whether  of  philosophy,  of  science,  or  of  life  in  general. 

A  subject  so  fundamental,  many-sided,  and  far-reach- 
ing will  naturally  detain  us  for  some  time.  The  wisdom 
we  seek  is  golden,  but  it  cannot  be  gained  by  any  of  the 
get-rich-quick  methods  characteristic  of  our  industrial 
and  neurasthenic  age;  the  way  to  it  is  a  little  long  and  I 
may  as  well  warn  you  that  in  these  lectures  I  intend  to 
pursue  It  in  a  leisurely  fashion.  The  study  is  not  so  "en- 
tertaining" as  a  "movie"  nor  so  easy  as  the  life  of  "mag- 
gots in  a  cheese"  or  that  of  summer  birds  in  a  valley  of 
fruits.  It  demands  some  patience,  hard  work  and  en- 
durance. It  will  quickly  weary  such  as  are  content  with 
a  little  phraseological  facility  in  matters  they  do  not 
understand,  but  not  those  whose  curiosity  is  deep  and 
genuine,  for  they  will  be  sustained  by  the  dignity  of  the 
task  and  the  joy  of  the  game. 

Let  us  now  enter  upon  it.  What  are  we  to  under- 
stand by  the  term  postulate?  You  are  aware  that  a 
branch  of  mathematics  (or,  for  that  matter,  of  mechanics 
or  of  physics  or  of  any  other  science),  If  the  branch  be 
ideally  constructed,  is  autonomous :  it  consists,  that  is,  of 
a  body  of  propositions  of  which  a  few  are  assumed — not 
proved  in  the  branch  but  taken  for  granted  there — and 
the  rest  are  deduced  from  them  as  logical  consequences. 
To  students  of  philosophy,  I  need  not  say  that  to  suppose 
all  the  propositions  of  an  autonomous  theory  to  be  proved 
in  it,  plainly  involves  circularity  and  a  contradiction  in 
terms.  In  accordance  with  current  usage,  which  I  Intend 
to  follow  in  this  matter,  any  proposition  thus  taken  for 
granted  in  a  given  branch  is  called  a  postulate,  or  assump- 
tion, or  axiom,  or  primitive  proposition,  or  fundamental 
hypothesis,  of  the  branch;  these  terms  being  used  inter- 


POSTULATES  41 

changeably  according  to  the  taste  of  the  author.  It  has 
not  always  been  so;  the  term  axiom,  for  example,  was 
long  used  to  denote  "self-evident  proposition,"  which  is  a 
kind  of  proposition  that  modern  mathematicians  have  not 
been  able  to  discover.  But  I  shall  not  detain  you  with 
an  historical  account  of  the  terms,  interesting  and  in- 
structive as  their  history  is.  It  gives  me  pleasure  to  say, 
however,  that,  if  you  feel  drawn  thereto,  as  I  hope  you 
do,  you  will  find  much  more  than  an  ample  clue  to  it  in 
the  introduction  to  Dr.  T.  L.  Heath's  superb  edition  of 
Euclid's  Elements  where  these  terms  and  kindred  matters 
are  set  in  the  bright  light  of  critical  commentary  from 
the  days  of  Plato  down  to  the  present  time.  In  passing, 
let  me  add,  by  way  of  indicating  an  opportunity,  that 
this  work  of  Dr.  Heath,  like  other  works  of  his,  attains 
a  high  degree  of  excellence  in  a  type  of  activity  in  which 
our  American  mathematical  scholarship  has  been  singu- 
larly lacking;  not  because  American  mathematicians  have 
lacked  facilities  or  ability,  for  these  they  have  not  lacked, 
but  because  the  universities  in  which  they  have  received 
their  training  and  have  done  their  work  have  not  yet  ac- 
quired the  requisite  atmosphere  and  spirit. 

A  postulate  is  one  thing;  a  system  of  postulates  is 
another.  In  defining  the  former,  I  have  by  no  means 
defined  the  latter.  It  is  not  easy  to  do  so  with  logical 
precision:  it  is,  I  mean,  not  easy  to  give  an  abstract 
definition  of  the  generic  concept  denoted  by  the  term, 
postulate  system;  and  I  shall  not  attempt  it  at  this  point, 
for  it  presupposes  study  of  the  concept  as  actually  re- 
vealed in  mathematics  and  so  has  its  proper  place  at  the 
end  of  the  study.  Here,  at  the  beginning,  we  must  be 
content  with  definition  by  example,  with  what  Professor 
Enriques,  in  his  Problems  of  Science,  has  called  concrete 


42  MATHEMATICAL   PHILOSOPHY 

definition,  which  is  nothing  more  mysterious  than  the  prac- 
tice, familiar  alike  in  science  and  in  ordinary  life,  of  tell- 
ing the  meaning  of  a  general  term  by  pointing  out  one  or 
more  of  the  many  objects  imperfectly  representing  it,  and 
saying,  "there,  there,  that  is  what  it  means,  or  that  and 
that."  I  wish  it  were  practicable  in  this  course  to  deal 
adequately  with  Definition  as  a  separate  topic,  with  its 
varieties,  its  functions  and  its  history.  It  is,  I  think,  an 
admirable  subject  for  a  scholarly  dissertation.  In  such 
an  undertaking  the  student  would  find  many  helpful  sug- 
gestions in  the  treatment  of  definition  by  Enriques  in  the 
work  just  now  mentioned;  in  certain  passages  of  Science 
and  Hypothesis  by  Poincare;  in  some  remarkably  keen 
observations  found  in  Pascal's  immortal  essays,  De  L'Es- 
■prit  Geometrique,  which  I  cited  in  the  preceding  lecture; 
in  the  above-mentioned  work  of  Dr.  Heath;  in  the  liter- 
ature of  symbolic  logic;  and,  as  I  need  not  say  to  you, 
who  are  students  of  philosophy,  in  the  Metaphysics  and 
Posterior  Analytics  of  Aristotle,  not  to  mention  the  Pla- 
tonic Dialogues  where  philosophy  in  our  western-world 
first  becomes  fully  conscious  that  the  way  to  wisdom — 
to  knowledge  of  things  eternal — is  not  the  way  of  song, 
however  glorious,  nor  that  of  sophistry,  however  preten- 
tious, but  is  the  way  of  logic,  and  where  accordingly,  de- 
spite the  presence  there  of  many  mystical  elements,  the 
spirit  of  Definition,  which  is  the  spirit  of  clear  thinking 
and  determinate  speech,  becomes  in  Socrates  a  conquering 
sword.  And  this  leads  me  to  say,  in  passing,  that  in  these 
our  democratic  times  of  free  speech  when  everyone,  no 
matter  how  ignorant  or  foolish,  is  a  licensed  prophet,  and 
blatant  sophists  abound  on  every  hand,  there  is  no  way 
in  which  you  as  teachers  of  philosophy  can  render  greater 
service  than  by  carrying  on  the  work  of  the  great  Gadfly 


POSTULATES  43 

— constraining  men  by  relentless  logical  criticism  to  a 
choice  of  one  or  the  other  of  two  alternatives:  coherency 
and  clarity  of  speech  or — silence.  Today,  the  mind  of 
the  world  is  a  weltering  sea  of  wild  passions  and  wilder 
opinions.  It  can  not  be  calmed  by  municipal  law,  but  it 
can  be  by  disciplining  men  to  a  decent  respect  for  the 
eternal  laws  of  thought.  And  that  is  the  supreme  obliga- 
tion of  philosophy  as  the  guardian  of  Reason. 

A  few  moments  ago  I  said  that,  in  the  beginning  of 
the  study  of  postulate  systems,  we  must  be  content  to 
define  the  notion  concretely — by  means,  that  is,  of  ex- 
amples. Accordingly,  I  am  going  to  spread  before  you 
presently  a  definite  system  of  postulates  and  invite  you  to 
examine  it  as  a  geologist  might  examine  a  specific  rock 
formation;  or  as  a  student  of  poetry  might  examine  a 
specific  poem;  or  a  student  of  law,  the  constitution  of  the 
Soviet  republic  or  that  of  the  United  States.  From  the 
large  variety  of  postulate  systems  recently  invented  for 
various  mathematical  branches,  I  have  selected,  as  a 
specimen  for  our  initial  study,  the  system  devised  by  the 
late  Professor  Hilbert  and  found  in  his  famous  Founda- 
tions of  Geometry.  It  is  one  of  several  systems  invented 
in  our  time  to  serve  as  logical  bases  of  Euclidean  Ge- 
ometry. Though  it  is  not  intrinsically  superior  to  its 
rivals,  whether  in  geometry  or  in  other  branches,  I  have 
selected  it  in  preference  to  them  for  two  reasons.  One 
of  them  is  that,  practical  arithmetic  not  being  a  science, 
Euclidean  Geometry  is  the  oldest  and  most  familiar 
branch  of  mathematics,  as  well  as  being  historically  the 
most  interesting  and  even  romantic. 

"The  clerk  Euclide  on  this  wyse  hit  fonde 
Thys  craft  of  gemetry  yn  Egypte  londe 


44  MATHEMATICAL   PHILOSOPHY 

In  Egypte  he  tawghte  hyt  ful  wyde, 
In  dyvers  londe  on  every  syde. 
Mony  erys  afterwarde  y  understonde 
Yer  that  the  craft  com  ynto  thys  londe. 
Thys  craft  com  into  England,  as  y  you  say, 
Yn  tyme  of  good  Kyng  Adelstone's  day." 

From  which  we  see  that  even  in  the  old  island  home 
of  our  beautiful  English  tongue  the  Greek  "Craft  of 
Gemetry"  has  been  known  for  a  thousand  years.  The 
second  reason  for  my  selecting  Hilbert's  system  is  that 
it  is  the  most  famous  of  all  existing  postulate  systems, 
save  one  only — that  of  Euclid.  Hilbert's  acquired  its 
great  fame  immediately,  not  entirely  by  its  merits,  for 
these,  as  already  said,  are  not  superior  to  the  merits  of 
some  other  systems,  but  largely  through  the  fame  of  its 
author,  which  was  world-wide.  If  you  ask  why  I  have 
chosen  it  instead  of  Euclid's  system,  which  surpasses  all 
others  in  fame,  the  answer  is  that,  though  Euclid's  system 
was  good  enough  to  withstand  more  than  two  thousand 
years  of  criticism,  it  is  now  known,  as  we  shall  see  later, 
to  have  some  grave  imperfections — most  of  them  sins 
of  omission.  The  postulates  of  Hilbert's  system  are 
called  axioms  by  him — "axioms  of  geometry."  As, 
however,  the  term  axiom  as  employed  by  him  is  exactly 
equivalent  to  the  term  postulate  as  I  have  defined  it,  I 
shall  be  doing  him  no  injustice  in  uniformly  referring  to 
his  system  as  a  system  of  postulates,  thus  avoiding  the 
term  axiom  as  likely  to  suggest  the  unavailable  notion  (so- 
called)  of  "self-evident  truth."  The  postulates  of  Hil- 
bert  fall  into  six  sets:  postulates  of  connection;  of  order; 
of  parallels;  of  congruence;  of  continuity;  of  complete- 
ness. I  give  them  as  found  in  the  authorized  English 
translation   of  Hilbert's  book  by  Professor  Townsend. 


POSTULATES  45 

Physically,  the  book,  as  you  observe,  is  small  and  light; 
but  spiritually  it  is  big  and  weighty.  Except  for  some 
harmless  abbreviations  of  statement,  the  postulates  to- 
gether with  the  definition  of  certain  terms  occurring  in 
them  are  as  follows: 

Postulates  of  Connection 

(i)   Two  distinct  points  determine  a  straight  line. 

(2)  Any  two  points  of  a  straight  line  determine  it. 

(3)  Three  non-collinear  points  determine  a  plane. 

(4)  Any  three  non-collinear  points  of  a  plane  de- 
termine it. 

(5)  If  two  points  of  a  line  are  in  a  plane,  every  point 
of  the  line  is  in  the  plane. 

(6)  If  two  planes  have  one  common  point,  they  have 
another. 

(7)  Every  straight  line  contains  at  least  two  points; 
every  plane  at  least  three  non-collinear  points;  and  space 
at  least  four  points  not  lying  in  a  plane. 

Postulates  of  Order 

(8)  If  A,  By  C  are  points  of  a  straight  line  and  B  is 
between  A  and  C,  then  B  is  between  C  and  A. 

(9)  If  A  and  C  are  two  points  of  a  straight  line,  there 
is  a  point  B  between  A  and  C,  and  a  point  D  such  that  C 
is  between  A  and  D. 

(10)  Of  any  three  collinear  points,  one,  and  but  one, 
is  between  the  other  two. 

(11)  Any  four  collinear  points,  A,  B,  C,  Z),  can  be  so 
arranged  that  B  shall  be  between  A  and  C  and  between 
A  and  D,  and  that  C  shall  be  between  A  and  D  and 
between  B  and  D. 


46  MATHEMATICAL   PHILOSOPHY 

Definitions. — A  pair  of  points,  A  and  B,  on  a  line, 
is  a  segment  AB  or  BA;  A  and  B  are  the  segment's  ends; 
the  points  between  A  and  5  are  the  segment's  points. 

(12)  Let  Ay  By  C  be  three  non-collinear  points  and 
let  a  be  a  line  of  their  plane  but  not  containing  any  of 
them.  If  a  contains  a  point  of  segment  AB,  it  contains 
a  point  of  segment  BC  or  of  segment  AC. 

Postulate  of  Parallels 

(13)  If  a  straight  line  a  and  a  point  A,  not  in  a,  be  in 
a  plane  a,  there  is  in  a  one  and  only  one  straight  line 
containing  A  but  no  point  of  a. 

Postulates  of  Congruence 

(14)  If  A  and  B  are  two  points  on  a  straight  line  a, 
and  if  a  point  A'  be  on  a  straight  line  a',  then  on  either 
side  of  A'  there  is  one  and  but  one  point  B'  such  that 
the  segment  AB  is  congruent  to  the  segment  A'B' . 
Every  segment  is  congruent  to  itself. 

(15)  If  a  segment  AB  is  congruent  to  a  segment 
A'B'  and  to  a  segment  A"B",  then  A'B'  is  congruent 
to  A"B". 

(16)  If  segments  y^i?  and  BC  of  a  straight  line  a  have 
no  common  point  but  B,  and  if  segments  A'B'  and  i?'C 
of  a  straight  line  a'  have  no  common  point  but  B',  then, 
if  AB  and  J5C  are  respectively  congruent  to  A'B'  and 
B'C,  AC  is  congruent  to  A"C". 

Definitions. — If  0  be  a  point  of  a  straight  line  <z, 
the  points  of  a  on  a  same  side  of  0  constitute  a  half-ray 
emanating  from  0;  a  pair  of  half-rays,  h  and  k,  emanating 
from  a  point  0  and  not  being  parts  of  a  same  straight 
line  is  an  angle  (h,  k);  0  is  the  angle's  vertex,  and  A  and  k 


POSTULATES  47 

its  sides;  its  interior  is  the  class  of  points  such  that,  if 
A  and  B  be  any  two  of  them,  segment  AB  contains  no 
point  of  h  or  k;  its  exterior  is  composed  of  all  other  points 
of  the  plane  except  0  and  the  points  of  h  and  k. 

(17)  Given  an  angle  (h,  k);  a  line  a!  in  a  plane  «;  a 
point  0  of  <z';  and  in  a  a  half-ray  h'  emanating  from  0; 
then  in  a  and  emanating  from  0  there  is  one  and  but  one 
half-ray  k'  such  that  the  angle  (h\  k')  is  congruent  to  (h,  k) 
and  that  the  interior  of  (h\  k')  is  on  a  given  side  of  <s'. 

(18)  If  the  angle  (h,  k)  is  congruent  to  (h'>  k')  and  to 
(A",  A"),  then  (A',  A')  is  congruent  to  (A",  A"). 

(19)  If,  in  the  triangles  ABC  and  ^'j&'C,  AB,  AC  and 
angle  ^y^C  are  respectively  congruent  to  A'B',  A'C  and 
angle  B'A'C,  then  the  angles  ABC  and  ^C5  are  respect- 
ively congruent  to  the  angle  A'B'C  and  A'C'B'. 

The  Postulate  of  Continuity  (or  of  Archimides) 

(20)  Let  the  point  A\  be  between  any  two  given 
points  A  and  B  of  a  straight  line  a.  Let  the  points 
A<z->  A$>  A 4,  ...  of  «2  be  such  that  A\  is  between  /^  and 
A2y  A2  is  between  A\  and  ^3,  and  so  on,  and  that  the 
segments  AAu  A\Ai<>  A2AZ,  .  .  .  are  mutually  congruent. 
Then  in  the  point  series  there  is  a  point  An  such  that  B 
is  between  A  and  An> 

Postulate  of  Completeness 

(21)  To  a  system  of  points,  lines  and  planes  it  is  not 
possible  to  add  other  elements  such  that  the  system  thus 
generalized  shall  form  a  new  geometry  in  which  all  the 
postulates  of  the  foregoing  five  sets  are  valid. 

Such  is  a  list  of  the  postulates  devised  by  Hilbert  to 
serve  as  a  foundation  of  Euclidean  geometry.     I  regret 


48  MATHEMATICAL   PHILOSOPHY 

having  had  to  detain  you  so  long  in  the  rather  arid  busi- 
ness of  presenting  so  long  a  list  in  detail.  My  apology 
is  the  importance  of  having  the  list  definitely  before  us. 
In  closing  this  lecture,  let  me  recommend  that,  as  a 
preparation  for  the  next  one,  you  familiarize  yourselves 
with  the  postulates  and  in  doing  so,  that  you  read  enough 
of  Hilbert's  book  to  see  how  carefully  the  theorems  are 
deduced  from  the  postulates  and  how  inevitably  they 
follow  therefrom. 


LECTURE  III 
Basic  Concepts 

PROPOSITIONAL     FUNCTION     AND     DOCTRINAL     FUNCTION 

MARRIAGE      OF      MATTER      AND      FORM ITS      INFINITE 

FERTILITY PROPOSITIONS    AND    DOCTRINES    THE    OFF- 
SPRING  VERIFIERS     AND     FALSIFIERS SIGNIFICANCE 

AND    NON-SENSE A    QUESTION    ASKED    BY    MANY    AND 

ANSWERED  BY  NONE. 

All  postulate  systems  have  certain  properties  or 
features  in  common.  In  connection  with  the  Hilbert 
system,  I  desire  to  draw  your  attention  to  such  of  these 
features  as  will  lead  us  to  form  a  certain  conception  which 
I  think  highly  important  and  to  which  I  have  given  the 
name — Doctrinal  Function. 

As  a  preliminary,  I  must  explain  briefly  a  closely  re- 
lated term — Propositional  Function — invented  by  Ber- 
trand  Russell;  it  is,  perhaps,  the  weightiest  term  that  has 
entered  the  nomenclature  of  logic,  or  mathematics,  in 
the  course  of  a  hundred  years.  It  has  the  rare  distinction 
of  being,  as  we  shall  see,  a  perfect  name  for  a  supreme 
concept.  Every  one  is  familiar  with  the  ordinary  notion 
of  a  function — with  the  notion,  that  is,  of  the  lawful 
dependence  of  one  or  more  variable  things  upon  other 
variable  things,  as  the  area  of  a  rectangle  upon  the  lengths 
of  its  sides,  as  the  distance  traveled  upon  the  rate  of 
going,  as  the  volume  of  a  gas  upon  temperature  and  pres- 

49 


50  MATHEMATICAL   PHILOSOPHY 

sure,  as  the  prosperity  of  a  throat  specialist  upon  the 
moisture  of  the  climate,  as  the  attraction  of  material 
particles  upon  their  distance  asunder,  as  prohibitionary 
zeal  upon  intellectual  distinction  and  moral  elevation, 
as  rate  of  chemical  change  upon  the  amount  or  the  mass 
of  the  substance  involved,  as  the  turbulence  of  labor 
upon  the  lust  of  capital,  and  so  on  and  on  without  end. 
This  familiar  notion  of  mutual  dependence  and  mutual 
variation  thus  exemplified  in  every  turn  and  feature  of 
life  and  the  world,  is  indeed  a  powerful  concept;  it  is, 
in  a  sense,  the  sole  subject  matter  of  science;  its  scien- 
tific name — function — was  first  pronounced,  it  is  said, 
by  Leibniz;  in  modern  mathematical  analysis,  it  has 
played  a  dominant  role,  giving  both  name  and  character 
to  certain  great  branches,  as  the  theory  of  functions  of 
real  variables  and  the  theory  of  functions  of  complex 
variables.  Yet,  powerful  as  it  is,  this  Leibnizian  con- 
ception, as  employed  in  traditional  mathematics,  is  far 
inferior  in  scope  to  that  denoted  by  prepositional  func- 
tion, which  indeed  embraces  the  former  as  a  special  case. 
What,  then,  are  we  to  understand  by  this  great  term? 

The  answer,  describing  rather  than  strictly  defining, 
is  that  a  propositional  function  is  any  statement  contain- 
ing one  or  more  real  variables,  where,  by  a  real  variable, 
is  meant  a  name  or  other  symbol  whose  meaning,  or 
value  as  we  say,  is  undetermined  in  the  statement  but 
to  which  we  can  at  will  assign  in  any  order  we  please  one 
or  more  values,  or  meanings,  now  one  and  now  another. 
I  fear  that  what  I  have  just  said  is  too  general  to  be  quite 
intelligible.  The  idea  can  be  made  sufficiently  clear, 
however,  by  some  simple  examples — by  concrete  defini- 
tion— provided  you  will  understand  that  the  examples 
are  to  the  general  concept  in  question  as  a  burning  match 


BASIC  CONCEITS  51 1 

to  a  world-conflagration  or  as  a  few  water  drops  to  a 
boundless  ocean.  If  we  denote  the  real  variables  by  such 
symbols  as  x,  y,  z,  w,  etc.,  then  for  simple  examples  of 
what  is  meant  by  propositional  function  we  may  cite  the 
following  quite  at  random:  #  is  a  man;  #  is  a  lover  of  y; 
x  is  the  specific  gravity  of  y;  x  is  a  noble  citizen  intemper- 
ately  desiring  to  impose  abstinence  on  y;  x  has  been 
divinely  appointed  by  y  to  subjugate  z;  2#  — 3y  =  ioz+a>; 
sin  x  =  cos  y;  x  denied  that  y  said  that  z  confessed  to  being 
the  author  of  w\  x  knows  that  y  voted  against  z  on 
account  of  jealousy  of  W\  and  so  on  ad  infinitum.  How 
many  variables  may  enter  a  propositional  function?  As 
many  as  we  please.  How  many  such  functions  are  there  ? 
Their  name  is  legion — the  host  of  them  is  literally  infinite. 
Even  so,  you  may  wish  to  say,  the  examples  are  not 
impressive.  Nevertheless,  the  concept  they  represent, 
each  in  its  little  way,  is  sovereign — "like  Jupiter  among 
the  Roman  gods,  first  without  a  second."1  Its  majesty, 
its  power,  its  subtlety,  the  immeasurable  depth  and  range 
of  its  significance  can  not  be  perceived  and  felt  at  once, 
but  only  more  and  more  with  days  and  months  and  years 
of  reflection.  You  will  reflect  upon  it  a  very  great  deal 
if  ever  you  enter  seriously  upon  the  study  of  symbolic 
logic. 

Let  us  reflect  a  little  upon  it  now.  There  will  be 
occasion  to  resume  its  consideration  at  a  later  stage.  At 
present,  I  wish  merely  to  direct  your  attention  to  the 
very  significant  fact  that  propositional  functions,  though 
they  have  the  forms  of  propositions,  are  not  propositions. 
It  is  of  the  utmost  importance  to  bear  that  in  mind.  A 
proposition  is  a  statement  that  is  true  or  else  false.  That 
is   why   propositions   are   so   important — they,   and   not 

1  Gladstone. 


52  MATHEMATICAL   PHILOSOPHY 

human  hearts,  are  the  residences — the  dwelling  places — 
of  those  curious  things  called  Truth  and  Falsehood.  A 
propositional  function,  owing  to  the  presence  in  it  of 
variables,  is  neither  true  nor  false.  The  statements 
2+7=9,  3+7=9>  are  propositions,  one  of  them  true, 
the  other  one  false;  but  the  statement,  x-{-y  =9,  is  neither 
true  nor  false;  it  is  not  a  proposition  but  is  a  propositional 
function. 

You  see  at  once  that  to  derive  propositions  from  a 
propositional  function  it  is  necessary  to  replace  the  latter's 
variables  with  what  we  may  call  constants,  or  values — 
with  terms  of  definite  meaning;  but  such  substitution, 
though  necessary,  is  not  sufficient,  for  it  is  always  pos- 
sible to  substitute  such  constants  as  will  give,  not  a 
proposition,  but  nonsense.  Suppose,  for  example,  that 
our  given  function  is  the  statement,  x  is  an  integer  less 
than  5.  Now,  the  class  of  all  integers  less  than  5  is  a 
constant — a  definite  somewhat.  Substituting  it  for  the 
variable  x,  we  get  the  statement,  the  class  of  all  integers 
less  than  5  is  an  integer  less  than  5.  This  statement  is 
neither  a  propositional  function  nor  a  proposition;  it  is 
nonsense — nonsense  consisting  in  talking  of  a  class  of 
things  as  if  a  given  class  could  conceivably  be  one  of  the 
things  composing  it;  as  if  the  class,  for  example,  of  loco- 
motives were  itself  a  locomotive;  or  as  if  the  class  of 
prohibitionary  moralists  were  itself  a  holy  constituent 
thereof;  or  as  if  the  class  of  apples  or  of  asses  were  itself 
an  apple  or  an  ass.  Such  "talking"  is  sheer  chattering, 
as  if  there  were  no  such  things  as  laws  of  Thought.  It 
is  evident  that  a  propositional  function  is  a  matrix  of  the 
propositions  derivable  from  it  by  substitution  and  has 
the  same  form  as  the  propositions  it  thus  moulds.  This 
latter  fact  should  be  noted  carefully  for  in  logic — that  is 


BASIC  CONCEPTS  53 

to  say,  in  mathematics — form  is  all-important — so  impor- 
tant indeed  that  some  critical  thinkers  have  ventured  to 
call  mathematics  the  science  of  Form. 

The  constants  that  convert  a  given  propositional  func- 
tion into  nonsense  may  be  called  inadmissible  constants 
for  that  function;  all  other  constants  may  be  called 
admissible  constants  for  the  function  since  they  convert 
it  into  propositions.  It  is  worthy  of  note,  in  passing, 
that  the  line  of  cleavage  between  the  admissible  and  the 
inadmissible  constants  for  a  given  function  is  not  always 
sharply  defined.  You  can  readily  construct  or  find  func- 
tions of  x  in  respect  of  which  it  may  be  doubtful  whether 
certain  constants — the  sweetness  of  sugar,  for  example, 
or  the  glory  of  renown — are  admissible  or  not.  You 
stand  here  before  an  open  and  inviting  field  for  research, 
the  problem  being  to  determine  criteria  for  deciding,  in 
the  case  of  any  propositional  function,  what  constants 
in  the  universe  of  constants  are  admissible  and  what  ones 
are  not.  The  situation  may  be  likened  to  that  of  physical 
organisms,  for  there  are  plants  and  there  are  animals,  but 
in  the  case  of  some  living  organisms  there  is  at  present 
no  means  of  deciding  to  which  division  of  the  kingdom 
they  belong. 

The  admissible  constants  for  a  given  function  fall  into 
two  classes:  those  converting  it  into  true  propositions  and 
those  converting  it  into  false  ones.  It  is  convenient  to 
call  the  constants  of  the  former  class  verifiers  of  the  func- 
tion; and  those  of  the  latter  class  falsifiers  of  it.  The 
verifiers  of  a  function  are  said  to  satisfy  it  and  are  called 
the  values  of  its  variables;  and  the  propositions  derived 
from  a  function  by  substituting  values  of  its  variables 
for  these  are  called  values  of  the  function.  Thus,  you 
see  that  a  propositional   function   is  itself  a  variable — 


54  MATHEMATICAL   PHILOSOPHY 

albeit  of  a  different  type  from  the  variables  it  contains — 
having  for  its  values  the  true  propositions  derivable  from 
it  by  means  of  its  verifiers. 

With  the  foregoing  ideas  and  distinctions  in  mind, 
iet  us  return  to  the  Hilbert  postulates  and  ask:  Are  they 
propositions  or  propositional  functions?  To  answer,  it  is 
necessary  and  sufficient  to  ascertain  whether  or  not  they 
contain  variables.  We  observe  at  once  the  presence  in 
them  of  certain  substantive  terms — "point,"  "straight 
line,"  "plane,"  and  "space" — which  seem  to  denote  the 
things  about  which  the  postulates  talk,  their  subject- 
matter — and  certain  relational  terms — "between"  and 
"congruent" — which  have  the  air  of  denoting  definite 
fundamental  relations  among  the  "points"  or  figures 
composed  of  them.  We  must  now  ask:  Do  these  terms 
denote  constants — things  of  unique  and  definite  meaning 
— or  do  they  play  the  role  of  variables?  Euclid  does  in- 
deed, as  you  know,  give  what  he  calls  "definitions"  of 
point,  line  and  plane,  but  in  his  proofs  and  constructions 
he  makes  no  use  whatever  of  the  so-called  definitions, 
which  he  ought  to  have  called  descriptions  designed  merely 
to  indicate  what  he  meant  by  the  terms;  or,  better,  he 
ought  to  have  omitted  the  definitions  as  logically  useless. 
As  to  the  term,  space,  it  does  not,  as  it  should  not,  occur 
in  Euclid's  Elements.  By  examining  Hilbert's  book,  you 
will  find  that  he  does  not  attempt  either  to  define  or  to 
describe  any  of  the  above-mentioned  six  terms,  except, 
of  course,  in  so  far  as  they  are  defined — restricted  in  their 
possible  meanings — by  having  to  satisfy,  or  verify,  the 
postulates.  The  omission  of  all  other  definition  of  them 
is  deliberate.  And  so  our  question  is  reduced  to  this: 
Does  the  requirement  that  the  things  denoted  by  the 
six  terms — "point,"  "straight  line,"  etc. — make  the  terms 


BASIC  CONCEPTS  55 

constants,  assign  to  each  of  them  a  unique  and  definite 
meaning?  The  answer  is  No:  each  of  the  terms  admits 
of  many,  infinitely  many,  different  definite  meanings 
satisfying  the  postulates.  The  answer  will  be  justified 
at  a  later  stage  of  our  discussion.  For  the  present,  I  ask 
you  to  assume  its  correctness.  We  may,  therefore,  now 
state,  in  answer  to  our  main  question,  that  the  six  terms 
are  not  constants,  but  variables,  and  that,  accordingly, 
the  postulates  are  not  propositions,  as  they  are  wont  to 
be  called,  but  are  propositional  functions.  As  you  reflect 
upon  this  fact,  you  will  find  that  its  importance  is  im- 
measurable, not  only  for  philosophy  in  its  narrower  sense, 
but  for  Criticism  1  in  the  widest  sense,  in  all  its  fields. 
In  a  future  lecture,  I  shall  return  to  the  matter  of  estimat- 
ing the  fact's  general  importance.  For  the  present,  let 
us  follow  its  strictly  logical  and  philosophical  leading. 

We  have  to  say  at  once  that  the  postulates  of  the 
system  we  are  examining  as  a  representative  specimen  of 
postulate  systems  in  general,  are  neither  true  nor  false, 
being  propositional  functions.  The  same  must,  of  course, 
be  said  of  all  the  theorems  deduced  or  deducible  from 
them  as  their  logical  consequences  or  implicates,  for  all 
such  theorems,  being  statements  involving  the  same 
variables  as  are  present  in  the  postulates,  are  propositional 
functions  and  are,  therefore,  neither  true  nor  false.  At 
this  point,  I  cannot  refrain  from  pausing  long  enough  to 
point  out  how  the  most  vitally  fundamental  fact  in 
logical  theory  appears  here  with  startling  vividness  in 
new  light.  Suppose  that  in  the  postulates  we  replace  the 
seven  terms — "  point,"  "  straight  line,"  "  plane,"  etc. — 
respectively,  by  any  meaningless  vocables  whatever,  as 

1  In  this  connection  the  reader  should  consult  Professor  F.  C.  S. 
Schiller's  very  suggestive  article  "Doctrinal  Functions"  in  The  Journal 
of  Philosophy,  Psychology  and  Scientific  Methods,  Vol.  XVI.,  1919. 


56  MATHEMATICAL   PHILOSOPHY 

loig,  boig,  ploigy  etc.,  so  that  postulates  (i)  and  (3),  for 
example,  shall  read:  (1)  Two  distinct  loigs  determine  a 
boig;  (3)  Any  three  loigs  not  in  a  same  boig  determine  a 
ploig.  Imagine  the  other  postulates  to  be  similarly 
restated.  Then,  of  course,  all  the  theorems  and  indeed 
the  entire  Hilbert  book  will  discourse  explicitly  about 
loigs,  boigs,  ploigs,  etc.,  and  nothing  else.  Do  not  fail 
to  note  now,  once  for  all,  that  as  thus  restated,  the 
theorems  and  postulates  are  related  precisely  as  before — 
the  former  being  logical  consequences  of  the  latter  and 
deducible  therefrom  without  even  the  slightest  change  in 
the  reasoning.  The  fact  which  thus  leaps  naked  into 
view  is  that  logical  deduction, — mathematical  demonstra- 
tion,-—<z//  valid  proof  in  no  matter  what  subject-matter, — 
depends  entirely  upon  the  form  of  the  premises,  or  pos- 
tulates, and  not  at  all  upon  any  specific  meanings  we 
may  assign  to  their  undefined,  or  variable,  terms  or 
symbols.  What  is  meant  by  propositional  form?  The 
question  has  been  often  asked  but  never  answered.  I 
ask  it  here  merely  to  signalize  its  importance.  It  is 
exceedingly  difficult.  I  hope  we  may  return  to  it  later. 
At  present,  let  us  go  on  with  the  central  thread  of  this 
lecture. 

We  have  seen  that  the  Hilbert  postulates  and  all  the 
theorems  logically  deducible  from  them  are  propositional 
functions.  So  important  a  fact  ought  not  to  be  con- 
cealed, not  even  from  the  physical  eye.  To  lay  it  bare, 
it  is  sufficient  to  replace  in  the  postulates  the  terms, 
there  playing  in  disguise  the  role  of  variables,  with  proper 
symbols  for  variables;  substituting,  let  us  say,  the 
symbols  vi,  V2,  V3,  V4,  respectively,  for  the  substantive, 
or  element-naming,  terms, — "  point,"  "  straight  line," 
"  plane,"  "  space," — and  for  the  relational  terms, — "  be- 


BASIC  CONCEPTS  57 

tween,"  "  congruent,"— the  symbols  i?i  and  R2.  Then 
postulate  (i)  will  read:  Two  distinct  »i's  determine  a 
v2.  For  another  example,  postulate  (8)  will  read:  If 
Vi,  vi'y  Vi"  are  v\ys  of  a  z>2  and  »i"  has  the  relation  Ri 
to  vi  and  t>i'"  then  »i"  has  Ri  to  t>i'"  and  v\.  It  is 
obvious  that  all  of  the  postulates  and  theorems  admit  of 
such  restatement.  I  strongly  recommend  that,  as  a  very 
enlightening  exercise,  you  thus  restate  all  of  the  postu- 
lates, a  few  of  the  theorems,  and  rewrite  the  proof  of  at 
least  one  of  the  latter. 

Having  thus  dragged  into  solar  light  the  fact, — hith- 
erto evident  only  in  the  psychic  light  of  understanding, — 
that  our  postulates  and  theorems  involve  variables,  let 
us  now  think  of  the  postulates  and  theorems  as  con- 
stituting a  Whole — a  definite  Body  of  logically  related 
propositional  functions.  Not  one  of  them  is  true;  not 
one  of  them  is  false.  What  is  true  is  that  the  postulates 
imply  the  theorems.  But  this  statement  of  implication, 
though  it  is  a  proposition  and  is  a  true  one — is  not  a  part 
of  the  Whole;  it  is  not  contained  in  the  Body  of  functions; 
were  we  to  put  it  in,  it  would  stand  there  alone  as  an 
intruder,  being  neither  one  of  the  postulates  nor  one  of 
the  theorems,  neither  a  premise  nor  a  conclusion,  neither 
an  implier  nor  an  implied;  it  is  a  philosophical  proposition 
about  the  Whole  but  is  not  a  member  of  it;  it  is  a  critical 
commentary  upon  it  but  not  upon  itself;  it  is  a  judg- 
ment,— a  just  and  important  judgment, — regarding  the 
Body  of  propositional  functions,  but  is  wholly  external 
to  it. 

This  definite  Body  of  logically  compendent  proposi- 
tional functions,  if  one  will  but  meditate  upon  it,  is  a 
truly  wonderful  thing — a  great  indestructible  shining 
Form  of  forms — "  poised    in  eternal  calm  "  above  the 


58  MATHEMATICAL   PHILOSOPHY 

changeful  things  of  the  world  of  sense.  What  shall  we 
call  it?  It  is  evidently  one  of  many,  for  every  postulate 
system  gives  rise  to  such  a  Form  and  many  of  these  sys- 
tems, as  we  shall  see,  are  essentially  different.  Shall  we 
call  it  Euclidean  Geometry  as  Hilbert  called  it  with  the 
world's  consent?  A  part  of  our  future  task  is  to  show 
that  it  has  neither  more  nor  less  to  do  with  geometry  as 
this  term  has  been  understood  from  time  immemorial 
than  with  a  thousand  other  things.  Shall  we  say  it  is  a 
Doctrine  of  a  certain  kind?  No;  for  a  doctrine  must 
have  a  specific  subject-matter,  which  our  Form  has  not; 
it  must  consist  of  propositions,  which  our  Form  does  not; 
it  must  be  true  or  else  false,  but  our  Form  is  neither. 

What,  then,  shall  we  say  it  is?  What,  pray,  ought 
our  Form, — our  definite  autonomous  Body  of  proposi- 
tional  functions, — to  be  called?  Observe  that  if  we 
replace  the  variables  in  its  postulated  functions  by  admis- 
sible constants,  we  thus  obtain  a  body  of  propositions 
matching,  in  one-to-one  fashion,  all  the  functions  of  our 
Body  of  functions;  we  thus  obtain,  that  is,  a  doctrine, 
for  the  body  of  propositions  has  a  specific  subject-matter 
and  is  true  or  false  according  as  the  substituted  con- 
stants are  all  of  them  verifiers,  or  some  of  them  falsifiers, 
of  the  postulated  functions.  Obviously,  we  may  thus 
obtain  various  doctrines  from  our  Body  of  functions  by 
substituting  various  sets  of  admissible  constants  for  the 
variables  in  the  postulated  functions.  It  is  obviously 
natural  to  call  the  true  doctrines  thus  derivable  the 
values  of  the  Body  of  functions. 

It  is  now  as  plain  as  the  noon-day  sun  what  the 
answer  to  our  question  must  be:  our  Body  of  logically 
related  propositional  functions,  since  it  is  a  thing  having 
doctrines    for   its   values    must    be   named    a    Doctrinal 


BASIC  CONCEPTS  59 

Function.  The  same  name  must,  of  course,  apply  to  the 
function  body  consisting  of  the  postulates  of  any  other 
postulate  system  together  with  the  theorems  logically 
deducible  from  them.  It  can  hardly  escape  your  atten- 
tion that  just  as  a  propositional  function  has  true  propo- 
sitions for  its  values,  a  doctrinal  function  has  true  doc- 
trines for  its  values;  that  just  as  we  viewed  a  proposi- 
tional function  as  the  matrix  of  all  the  propositions  (true 
or  false)  derivable  from  it  by  substitution  of  admissible 
constants,  so  we  may  view  a  doctrinal  function  as  the 
matrix  of  all  the  doctrines  (true  or  false)  derivable  from 
it  in  like  manner;  and  that  just  as  a  given  propositional 
function  and  the  propositions  derivable  from  it  are  iden- 
tical in  form,  so  a  given  doctrinal  function  and  the 
doctrines  derivable  from  it  are  the  same  in  respect  of 
form;  they  are  isomorphic,  as  we  say.  In  marriage  with 
subject-matter,  a  Doctrinal  Function  becomes  the  matrix 
of  an  infinite  family  of  doctrines;  the  children  inherit  the 
form  of  the  mother. 

It  will  be  convenient  to  say  that  we  are  interpreting 
a  given  doctrinal  function  whenever  we  derive  from  it, 
in  the  way  now  familiar,  one  of  its  values,  or  true  doc- 
trines; and  these  values,  or  true  doctrines,  may  be  con- 
veniently called  interpretations  of  the  function. 


LECTURE  IV 
Doctrinal  Interpretations 

A     MOTHER     OF     DOCTRINES     MISTAKEN     FOR     HER     ELDEST 

CHILD INFINITELY    MANY    INTERPRETATIONS    OF    ONE 

DOCTRINAL  FUNCTION ORDINARY  GEOMETRY  BUT  ONE 

OF       THEM OTHER       INTERPRETATIONS       GEOMETRIC, 

ALGEBRAIC  AND  MIXED — IDENTITY  OF  FORM  WITH 
DIVERSITY  OF  CONTENT — DISTINCTION  OF  LOGICAL 
AND      PSYCHOLOGICAL — PROJECTIVE     GEOMETRY     THE 

CHILD    OF    ARCHITECTURE A    SCIENCE     BORN    OF    AN 

ART — INFINITE  POINTS  AND  THE  MEETING  OF  PARAL- 
LELS  POLE-TO-POLAR      TRANSFORMATIONS LOGICAL 

USE  OF  PATHOLOGICAL  CONFIGURATIONS. 

In  the  following  discussion,  I  shall  assume  that  you 
have  before  you  the  Hilbert  postulates  as  restated  in 
terms  of  the  variable-symbols,  v\>  V2,  V3,  v*,  Ri  and  R2. 
It  will  be  convenient  to  call  the  doctrinal  function  con- 
sisting of  these  postulates  and  their  consequent  theorems 
the  "  Hilbert  doctrinal  function  "  and  to  denote  it  by 
HaF'  .  Now  be  good  enough  to  note  very  carefully  that, 
if  we  omit  from  the  postulates  all  reference  to  points  not 
in  a  given  plane,  the  remaining  postulates  together 
with  their  theorematic  consequences  constitute  another 
doctrinal  function  and  that  this  is  included  in  HaF'. 
Let  us  denote  the  minor  function  by  HaF.  The 
purpose  of  this  lecture  is  to  present  or  rather  to  indi- 

60 


DOCTRINAL  INTERPRETATIONS  61 

cate  some  of  the  infinitely  many  values,  or  interpreta- 
tions, of  these  two  functions;  to  indicate,  that  is,  some 
of  the  true  doctrines  having  the  functions  for  their 
common  mould. 

One  of  the  interpretations  of  HaF'  is  the  familiar 
doctrine  which  results  from  letting  the  symbols,  v\,  z% 
z>3,  V4,  Ru  R2,  denote,  respectively,  point,  straight  line, 
plane,  space,  between  and  congruent,  or  equal,  taken  in  the 
sense  in  which  they  have  been  taken  from  pre-Euclidean 
days, — in  the  sense  in  which  they  (or  some  of  them) 
are  "described  "  by  Euclid  in  the  Elements, — in  the  sense 
in  which  Hilbert  takes  them  in  his  Foundations  as  shown 
by  the  drawings  or  figures  he  there  employs  and  which  is 
doubtless  responsible  for  his  calling  his  book  The  Founda- 
tions of  Geometry.  This  special  interpretation  of  HaF', — 
this  special  value  of  that  function, — this  special  doctrine, 
which  I  shall  denote  by  D\, — is,  you  observe,  the  ordi- 
nary Euclidean  Solid  Geometry,  or  geometry  of  three 
dimensions,  with  which  we  all  of  us  gained  some  acquaint- 
ance in  high  school  or  college  despite  the  somewhat  rough 
or  uncritical  way  in  which  it  was  there  presented  as  for 
beginners.  The  corresponding  interpretation  of  HaF  is 
the  yet  more  familiar  Euclidean  geometry  of  the  plane,  a 
two-dimensional  geometry.  Denote  it  by  D\.  I  shall  take 
both  D\  and  D\  for  granted,  assuming  them,  whenever 
it  is  convenient  to  do  so,  in  future  discussion. 

Let  me  now  direct  your  attention  to  another  geometric 
interpretation  of  the  two  functions— to  one  which,  though 
it  is  near-lying  and  fairly  obvious,  has  not,  so  far  as  I  am 
informed,  been  published.  In  order  to  present  it  intel- 
ligibly, I  must,  by  way  of  preparation,  make  you  ac- 
quainted with  the  concepts  of  projective  straight  line, 
projective  plane  and   projective  space,   for,   as  you  will 


62  MATHEMATICAL   PHILOSOPHY 

recall,  I  have  not  assumed  on  your  part  a  knowledge  of 
Projective  Geometry.  It  will  be  sufficient  for  our  pur- 
pose to  introduce  them  in  the  rough  traditional  way 
instead  of  the  very  refined  way  employed  by  Veblen  and 
Young,  for  example,  in  their  Projective  Geometry,  which  is 
based  upon  a  postulate  system  appropriate  for  projective 
geometry. 


Fig.  i. 

Let  the  figure  be  in  a  Euclidean  plane — the  kind  of 
plane  belonging  to  D\.  All  lines  of  the  plane  that  con- 
tain a  given  point  P  constitute  a  pencil  of  lines;  P  is  the 
pencil's  vertex.  All  the  points  of  a  line  L  constitute  a 
range  of  points;  L  is  the  range's  base.  It  is  plain  that 
each  point  of  range  L  is  on  one  line  of  pencil  P;  and 
that,  reciprocally,  each  line  of  P  has  one  point  of  L,  with 
a  single  exception, — L\  parallel  to  L,  contains  no  point 
of  L.  To  remove  this  exception  to  the  one-to-one  corre- 
spondence, otherwise  perfect,  there  is  made  in  projective 
geometry  an  agreement  or  convention:  namely,  that 
each  line  has  (at  an  infinite  distance)  a  so-called  "  ideal  " 
point,  or  point  at  infinity,  and  that  the  "  ideal  "  points 
of  any  two  parallel  lines  are  coincident.  We  thus  get, 
as  you  see,  a  new  sort  of  straight  line  and  of  plane  and  of 


DOCTRINAL  INTERPRETATIONS  63 

space,  which  we  describe  by  calling  them  respectively 
projective  straight  line,  projective  plane  and  projective 
space.  The  adjective  has  fine  propriety,  but  that  need 
not  here  detain  us.  You  can  readily  prove,  or  you  may 
assume,  that  the  "  ideal  "  points  of  the  projective  plane 
constitute  a  straight  line — called  the  "  ideal  "  line,  or 
line  at  infinity;  and  that  the  locus  of  the  "  ideal  "  points 
of  projective  space  is  a  plane — called  the  "  ideal  "  plane, 
or  plane  at  infinity.  I  can  not  pause  here  to  justify  the 
convention.  It  is  amply  justified  by  its  consequences, 
for  which,  if  you  be  interested,  you  must  repair  to  pro- 
jective geometry, — invented  by  the  engineer,  Desargues, 
a  contemporary  of  Descartes  and  Pascal, — quickly  for- 
gotten— reinvented,  in  France  again,  about  one  hundred 
years  ago — perhaps  the  most  beautiful  branch  of  mathe- 
matics. 

We  may  now  proceed  to  the  promised  new  interpre- 
tation of  our  doctrinal  functions.  As  HaF  is  simpler 
than  HaF',  let  us  first  deal  with  the  former. 

Let  ir  denote  a  projective  plane.  Let  a  chosen  point 
0  be  the  vertex  of  a  pencil  of  lines  of  71-;  call  each  line  of  the 
pencil  an  (9-line.  Note  that  every  other  pencil  of  r 
contains  one  and  but  one  (9-line.  Now  let  us  in  thought 
remove  from  71-,  once  for  all,  the  (9-pencil.  We  thus 
remove  one  and  but  one  line  from  every  other  pencil. 
We  may  conveniently  call  the  pencils,  thus  bereft  of  a 
line,  pathopencils  as  being  defective  or,  so  to  speak, 
pathological.  We  have  taken  from  -w  one  and  but  one 
pencil  of  lines.  Our  field  of  operation  consists  of  all  that 
is  left.  Denote  the  field  by  <l>.  We  are  going  to  give 
HaF  an  interpretation  in  $;  the  interpretation,  as  you 
will  see,  will  be  a  doctrine  about  certain  things  in  $ — 
a  geometry  of  the  field.     The  interpretation  results  from 


64 


MATHEMATICAL   PHILOSOPHY 


assigning  to  the  vs  and  the  R's  in  the  postulates  otllAF  the 
following  meanings,  or  constant  values:  v\  is  to  mean  a 
line  of  $;  02,  a  pathopencil  of  <£;  i?i  is  to  mean  "  between  " 
in  the  sense  that,  if  A,  B,  C  be  three  lines  of  a  pathopencil, 
B  will  be  considered  to  be  between  A  and  C,  if  A  (or  C) 
must  rotate  through  the  position  of  B  to  coincide  with  C 
(or  A)  (for,  of  course,  a  line  of  a  pathopencil  must  not 
be  supposed  to  rotate  into  the  position  left  vacant  by 
the  absent  0-line);  and  R2  is  to  mean  "  congruent  "  in  a 
sense  to  be  given  later. 


Fig.  2. 


We  have  to  show  that  the  indicated  meanings  satisfy, 
or  verify,  the  postulates  of  HaF.  That  some  of  them  are 
thus  satisfied  may  be  made  evident  by  simple  figures; 
and  it  will  be  interesting  and  enlightening  to  exhibit 
such  evidence  before  giving  the  proof  for  all  the  postulates. 
At  the  same  time,  we  will  lay  bare,  by  means  of  figures, 
the  significance  of  one  or  two  theorems  of  the  new  doc- 
trine. I  shall  not  here  repeat  the  postulates,  but  will 
suppose  you  to  have  them  in  hand. 

Postulate  (1)  is  plainly  satisfied,  for  any  two  lines 
A  and  B  of  <£  determine,  as  in  Fig.  2,  a  pathopencil  a, 
which  consists  of  all  the  lines  through  a  except  the  0-line 
Oa. 


DOCTRINAL  INTERPRETATIONS 


65 


Next  consider  postulate  (8).  That  it  is  verified  is 
evident  in  Fig.  3  where  line  B  is  clearly  between  A  and  C 
and  between  C  and  A.     For  another  example,  let  us  take 


Fig.  3. 

postulate  (12),  the  famous  postulate  of  Pasch.     But  first 
we  must  have  some 

Definitions. — A  pair  of  lines,  A  and  B,  of  a  patho- 
pencil,  is  a  segment  AB  or  BA;  its  ends  are  A  and  B;  the 
lines  between  them  are  the  segment's  lines. 


Fig.  4. 

In  the  light  of  Fig.  4,  it  is  obvious  that  postulate  (12) 
is  satisfied.  Note  that  A,  B,  C  are  any  three  lines  of  «J> 
not  belonging  to  a  same  pathopencil;  that  pathopencil  a 
contains  a  line  of  segment  AB,  by  hypothesis;  and  that  a 
contains  a  line  of  segment  BC  but  none  of  segment  AC. 


66 


MATHEMATICAL   PHILOSOPHY 


Before  considering  another  postulate,  let  us  illustrate 
the  following  theorem  (a  propositional  function  in  the  doc- 
trinal function  HaF):  Any  given  z>2  separates  the  remaining 
v\s  of  the  V3  into  two  classes  such  that,  if  v\  and  v\"  are  one 
of  them  in  one  of  the  classes  and  the  other  in  the  other,  the 
segment  v\  v\"  contains  a  v\  of  the  V2;  and  that,  if  v\  and 
v\"  are  both  in  one  of  the  classes,  the  segment  does  not 
contain  a  v\  of  the  v2.  (It  is  theorem  5  of  Hilbert's  book.) 
A  fairly  careful  examination  of  Fig.  5  will  suffice  to  con- 


Fig.  5. 

vince  you  that  that  theorem  is  verified  in  our  new  inter- 
pretation. One  of  the  two  classes  of  lines  is  composed 
of  all  the  lines  of  $>  that  go  between  0  and  a;  all  the 
other  lines  compose  the  other  class.  Segments  A  A'  and 
BB'  contain  no  line  of  the  pathopencil  a,  but  any  such 
segment  as  AB  contains  a  line  of  a. 

You  should  not  fail  to  compare  Fig.  5  with  Hilbert's 
figure  for  the  corresponding  proposition  in  doctrine  D\, 
the  old  familiar  interpretation.  The  two  figures  are  the 
same  logically  but  very  different  psychologically:   in  the 


DOCTRINAL  INTERPRETATIONS  67 

latter  figure  the  truth  of  the  proposition  is  perfectly  and 
immediately  evident  to  intuition,  while  in  the  former  the 
truth  of  the  proposition  is  very  far  from  being  thus  evi- 
dent. Why?  The  question,  you  observe,  is  one  for 
psychologists,  like  hundreds  of  similar  questions  that 
arise  here  and  elsewhere  in  mathematics,  if  only  psycholo- 
gists would  learn  enough  mathematics  even  to  ask  the 
questions. 


Fig.  6. 

Let  us  now  turn  to  postulate  (13) — the  postulate  of 
parallels.  Fig.  6  shows  clearly  that  this  famous  Euclidean 
postulate  is  satisfied  by  our  new  interpretation.  Here  a 
is  the  given  pathopencil;  A  is  any  given  line  not  belonging 
to  a;  b  is  a  pathopencil  containing  A  but  having  no  line 
in  common  with  a,  and  there  is  plainly  no  other  such 
pathopencil;  in  other  words,  b  is  parallel  to  a  and  there 
is  no  other  such  pathopencil  containing  A. 

If,  now,  you  attempt  to  show  (and  I  advise  you  to 
make  the  attempt)  by  a  figure  that  postulate  (20),  or 


68 


MATHEMATICAL   PHILOSOPHY 


other  postulate  involving  congruence,  is  satisfied,  using 
"  congruent  "  in  your  figure  in  the  sense  it  has  in  the  old 
interpretation  or  doctrine  £>i,  you  will  quickly  find 
yourselves  in  trouble.  In  the  new  interpretation,  how- 
ever, we  are  not  going  to  employ  "  congruent  "  in  that 
sense,  but  in  a  sense  which  I  shall  explain  presently  in  the 
course  of  a  simple  argument  designed  to  show,  as  by  a 


Fig.  7. 

single  stroke,  that  all  of  the  postulates  are  satisfied  by  our 
new  interpretation. 

Before  presenting  that  argument  we  must  acquaint 
ourselves  with  what  is  called,  in  the  projective  geometry 
of  a  plane,  the  Pole-Polar  transformation  with  respect  to  a 
circle.  It  is  a  very  beautiful  transformation,  important, 
and  easy  to  understand. 

Let  Fig.  7  be  in  a  projective  plane  t.  Tangents 
through  P  are  drawn  to  the  circle.  Line  L  joining  the 
points  of  tangency  is  called  the  polar  of  P,  which  is  called 


DOCTRINAL  INTERPRETATIONS  69 

the  pole  of  L.  You  readily  see  that,  if  P  moves  off, 
becoming  an  "  Ideal  "  point  of  tt,  the  polar  L  goes  through 
the  center — is  a  line  of  the  pencil  vertexed  at  0;  also,  if 
P  moves  up  to  the  circle,  L  becomes  tangent  at  P.  If  P 
is  inside  the  circle,  say  at  P',  L  is  Z/,  whose  construction 
is  obvious;  if,  in  particular,  P  is  at  0,  L  is  x's  "  ideal  " 
line,  or  line  at  infinity.  Thus  you  see  that  the  given  circle 
serves  to  set  up  a  one-to-one  correspondence  between  the 
points  of  ir  as  poles  and  the  lines  of  r  as  polars.  This 
correspondence  is  called  the  Pole-Polar  transformation  of 
ir  with  respect  to  the  circle.  We  say  the  transformation 
transforms  or  converts  a  point  into  its  polar  line,  a  line 
into  its  pole  point,  and  each  of  these  is  called  the  transform 
of  the  other.  If  you  will  study  the  transformation  a  bit, 
playing  with  it,  making  a  few  figures,  you  will  discover 
some  of  its  important  properties,  such  as  these:  it  con- 
verts a  range  of  points  into  a  pencil  of  lines,  and  a  pencil 
into  a  range;  a  segment  of  a  range  into  a  segment  of  a 
pencil,  and  a  pencil  segment  into  a  range  segment;  if 
three  points  of  a  range  or  three  lines  of  a  pancil  are  in  the 
order — A,  B,  C, — the  transforms  are  in  the  same  order. 

And  now  for  the  argument  showing  that  all  the  pos- 
tulates in  HaF  are  verified  by  our  new  interpretation. 
Imagine  our  field  <J>  laid  down  upon  a  Euclidean  plane  a. 
Remember  that  the  0-pencil  is  not  in  $ — I  have  put  in  a 
few  of  its  lines  merely  to  remind  us  that  it  is  absent. 
Such  a  pencil  is  present  in  a  just  below.  Remember  also 
that  4>  has  an  "  ideal  "  line  at  infinity  which  a  has  not. 
Assume  a  definite  circle  C  about  0  as  center.  Consider 
the  pole-polar  transformation  as  to  C.  Let  the  trans- 
forms of  the  points  and  lines  of  a  be  in  <l>;  you  readily 
see  that,  in  a  one-to-one  way,  the  points  of  a  are  converted 
into  the  lines  of  $  and  the  lines  (ranges)  of  a  into  the 


70 


MATHEMATICAL   PHILOSOPHY 


pathopencils  of  <£;  also  that  the  order  of  the  elements  in 
a  is  carried  over  into  their  transforms  in  <$.  But,  as  you 
readily  see,  congruence  in  a, — that  is,  congruence  as  under- 
stood in  interpretation  Z>i, — is  not  carried  over.  We, 
therefore,  agree  to  give  a  new  meaning  to  "  congruent  " 
for  use  in  $,  and  the  meaning  is  this :  if  two  segments  or 
angles  be  congruent  (in  the  old  sense)  in  a,  then  and  only 


Fig.  8. 


then  their  transforms  shall  be  said  to  be  congruent  in  <f>. 
It  is  evident,  without  further  talk,  that  all  the  postulates 
are  satisfied  and  that  we,  accordingly,  have  a  new  inter- 
pretation of  the  doctrinal  function  HaF.  Let  us  denote 
this  interpretation,  or  doctrine,  by  D2.  D2  is  evidently  a 
two-dimensional  geometry  of  the  lines  and  pathopencils 
of  <£  and  is  isomorphic  with  Di,  the  ordinary  geometry  of 
the  points  and  lines  of  a  Euclidean  plane. 


DOCTRINAL  INTERPRETATIONS  71 

I  will  close  this  lecture  by  indicating, — merely  indicat- 
ing,— the  analogous  new  interpretation  of  the  Doctrinal 
function  HaF',  which,  you  remember,  includes  the  entire 
list  of  Hilbert  postulates  in  their  restated  form.  I  shall 
denote  the  new  doctrine,  or  interpretation,  by  ZV.  Let 
S  denote  a  projective  space  of  three  dimensions.  We  have 
already  formed  the  concept  of  such  a  space.  All  the  lines 
(or  planes)  of  S  that  have  in  common  point  P  are  together 
called  a  sheaf,  or  bundle,  of  lines  (or  planes) ;  all  the  planes 
having  a  common  line  constitute  an  axial  pencil  of  planes. 
Let  0  be  a  chosen  point  of  S.  Call  the  sheaf  of  lines  (or 
planes)  having  0  for  vertex  the  (9-sheaf  of  lines  (or  planes). 
In  thought  remove  from  S  the  (9-sheaf  of  lines  and  the 
(9-sheaf  of  planes.  We  thus  remove  from  every  other 
line  sheaf  one  line,  from  every  other  plane  sheaf  an  axial 
pencil  and  from  every  axial  pencil  (not  contained  in  the 
(9-sheaf  of  planes)  one  plane.  The  ensembles,  thus  ren- 
dered defective,  may  be  respectively  called  a  pathosheaf 
of  lines,  a  pathosheaf  of  planes  and  a  pathopencil  of 
planes,  or  plane  pathopencil.  Analogous  to  the  pole- 
polar  transformation  as  to  a  circle, — which  we  have 
already  explained  and  used, — there  is  for  S  a  pole-polar 
transformation  with  respect  to  any  given  sphere  convert- 
ing each  point  into  a  polar  plane  and  each  plane  into  pole 
point.  Our  field  of  operation — $' — is  S  bereft  of  the  two 
O-sheaves.  As  you  may  have  by  this  time  surmised,  our 
new  interpretation,  or  doctrine  ZV \  arises  on  giving  the 
variable  symbols  in  the  postulates  of  HaF'  meanings  as 
follows:  v\  will  mean  a  plane  of  3>';  V2,  a  pathopencil  of 
planes;  z%  a  pathosheaf  of  planes;  V4,  $';  Ri,  between 
in  the  sense  that,  if  A,  B,  C  are  planes  of  a  pathopencil, 
B  will  be  said  to  be  between  A  and  C  if  either  of  the  latter 
must  rotate  through  the  position  of  B  to  coincide  with 


72  MATHEMATICAL   PHILOSOPHY 

the  other;  Ri  will  mean  congruent  in  the  sense  that 
segments,  etc.,  in  <£'  will  be  called  congruent  if  they  are 
transforms  of  segments,  etc.,  congruent  in  Dz- 

Obviously  D2  is  a  three-dimensional  geometry  of 
planes,  pathopencils  of  planes  and  pathosheaves  of  planes 
of  $'  and  is  isomorphic  with  D\\  the  familiar  geometry 
of  points,  lines  and  planes  of  ordinary  Euclidean  space. 

Note  that  D2  and  D2  are  logically  the  same  as  D\  and 
D\  but  greatly  differ  from  the  latter  psychologically. 


LECTURE  V 
Another  Geometric  Interpretation 

BRIEF  INTRODUCTION  TO  THE  METHOD  OF  DESCARTES  AND 

FERMAT INVERSION        GEOMETRY       AND        INVERSION 

TRANSFORMATION — THE    INFINITE    REGION    OF    INVER- 
SION    SPACE     A     POINT BUNDLES     OF     CIRCLES     AND 

CLUSTERS    OF     SPHERES PATHOCIRCLES    AND    PATHO- 

SPHERES ONE-TO-ONE  CORRELATION. 

In  presenting  a  third  interpretation  of  our  two  doc- 
trinal functions,  it  will  be  convenient  to  borrow  a  few 
ideas  from  Cartesian  Analytical  Geometry  and  Inversion 
Geometry.  It  will  be  advantageous  to  explain  them  in 
advance. 

The  perpendicular  lines  OX  and  0Y>  Fig.  9,  are  called 
coordinate  axes;  0  is  the  origin  of  distances,  which,  if 
measured  upward  or  rightward,  are  regarded  positive,  but, 
if  downward  or  leftward,  negative.  I  am  supposing  the 
figure  to  be  in  a  Euclidean  plane.  Choose  some  unit  of 
length;  then  any  point  has  a  pair  of  numbers  (x,  y)y 
P's  distances  from  the  axes  and  called  its  coordinates. 
Conversely,  to  any  such  a  pair  belongs  a  point.  Let  (1), 
Fig.  10,  be  any  line  through  0;  then  (2),  parallel  to  (1), 
is  any  line  of  the  plane.  Let  P(x,  y)  be  any  point  of  (1); 
let  ra=tan  0;  then  y=mx;  this  equation  is  the  equation 
of  (1);  it  is  so  called  because  to  any  pair  (x,  y)  satisfying 
it  belongs  a  P  of  (1)  and  any  P  of  (1)  has  a  pair  satisfying 

73 


74 


MATHEMATICAL   PHILOSOPHY 


it.     Plainly,  the  y  of  P'  is  equal  to  P's  y+b;    hence  the 
equation  of  (2),   any  line  of  the  plane,  is:    y  =  mx-\-b. 

Y 


<ZhD 


Fig.  9. 


Fig.  10. 


Conversely,  any  equation  of  first  degree  in  x  and  y  repre- 
sents a  line  of  the  plane. 


ANOTHER  GEOMETRIC   INTREPRETATION      75 


By  Fig.  II  you  see  that,  if  d  is  the  distance  between 
two  points,  Pi(#i,  y\)  and  ^2(^2,  y 2),  then  d2  =  {x\  —  x2)2 
+  (yi-y2)2. 

From  the  foregoing  distance  formula,  you  see  that  the 
equation  of  any  circle,  Fig.  12,  of  radius  r  and  center 
{a,  b)  is  (x  —  a)2  +  (y  —  b)2  =  r2 ;  that  is,  #2  +y2  —  lax  —  2by  + 
a2+lP—  r2=0.  Conversely,  any  equation  of  the  form 
x2-\-y2+2dx+2By+C  =  0  represents  a  circle  of  center 
(  —  Ay  —B)  and  squared  radius,  A2-\-B2  —  C. 


Fig.  ii. 

On  any  line  through  the  center  of  a  circle  of  radius  r 
let  P,  P'  be  such  that  distance  OP  times  distance  OP'  =r2; 
the  point  P  (or  P')  is  called  the  inverse  of  P'  (or  P) ;  the 
circle  and  its  center  are  called  the  inversion  circle  and 
center.  Taking  the  circle's  center  for  origin,  Fig.  13,  you 
will  easily  find  that: 


(0 


x  r 


't2 


x  = 


*'2+y'2' 


y  = 


y'r2 


(2) 


*'2+y' 


x  — 


76 


MATHEMATICAL   PHILOSOPHY 


Notice  that  to  each  point  there  corresponds  one  and  but 
one  point — except  that  the  inversion  center  corresponds 
to  no  point  (in  the  Euclidean  plane).  To  remove  this 
exception  it  is  common  to  assume  the  existence  of  one 
and  but  one  "  ideal  "  point,  or  point  at  infinity,  to  serve 


P(x.yJ 


-e 


Fig.  12. 


as  the  inverse  of  the  center.  The  new  sort  of  plane  thus 
got  is  called  the  Inversion  Plane.  The  foregoing  point-to- 
point  correspondence  is  called  the  Inversion  Transforma- 
tion of  the  plane  with  respect  to  the  given  circle.  Clearly, 
any    line    through    the    center    is    converted    into    itself. 


ANOTHER   GEOMETRIC   INTERPRETATION       77 

What  is  the  transform,  or  inverse,  of  a  line  not  through 
the  center?  Let  Jx-\-By-\-C  =  0  be  such  a  line;  replace 
the  coordinates  (x,  y)  of  any  point  in  it  by  their  values 
taken  from  (i),  simplify  the  result  and  (if  you  like)  drop 
the  primes;  we  thus  get 

A  B 

(3)  x2+y2+r2—  x+r2-y=0. 


Fig.  13. 

This,  you  note,  is  a  circle  through  the  inversion  center, 
which  is  here  the  origin,  for  the  coordinates  (0,  0)  of  the 
origin  satisfy  the  equation.  Hence  every  line  not  through 
the  center  has  for  its  transform,  or  inverse,  a  circle  through 
the  inversion  center. 

With  these  simple  ideas  held  in  reserve  for  use  as  we 
need  them,  let  us  proceed  to  our  third  geometric  interpre- 
tation. It  will  be  advantageous  to  deal  first  with  IlAF. 
Denote  by  -k  an  inversion  plane.  Let  0  be  a  chosen  point 
of  it.     The  ensemble  of  all  circles  through  0  is  called  a 


78  MATHEMATICAL   PHILOSOPHY 

bundle  of  circles.  The  bundle  includes,  as  infinite  circles 
(i.e.,  circles  of  infinite  radius),  the  straight  lines  through 
0.  Now,  in  thought,  let  us,  once  for  all,  remove  the 
point  0  from  ir.  Each  circle  of  the  bundle  now  lacks  a 
point;  we  may  call  them  pathocircles,  and  speak  of  the 
(9-bundle  of  pathocircles.  Our  field  of  operation — which 
may  be  denoted  by  K — is  composed  of  the  patho- 
circles of  the  0-bundle  and  the  points  (except  0, 
of  course)  of  t.  We  are  going  to  give  the  doctrinal 
function  HaF  an  interpretation  in  the  field  K;  it  will  be 
a  geometry  of  certain  elements  of  K.  The  interpretation 
arises  from  assigning  to  the  variable-symbols  in  the 
postulates  of  HaF  definite  meanings  as  follows:  vi  will 
mean  a  point  of  K\  V2,  a  pathocircle;  Ri  will  mean 
between  in  the  sense  that,  if  A,  B,  C  be  three  points  of  a 
pathocircle,  B  will  be  said  to  be  between  A  and  C,  if  A 
(or  C)  must  go  through  B  in  moving  on  the  pathocircle 
to  C  (or  A);  and  .#2  will  mean  congruent  in  the  sense 
that,  if  two  segments  or  angles  be  congruent  in  the 
ordinary  sense  (interpretation  Di),  their  transforms,  or 
inverses,  with  respect  to  a  given  circle  with  0  as  center, 
will  be  called  congruent  in  the  field  of  K. 

We  have  now  to  show  that  the  postulates  are  verified 
by  the  meanings  assigned.  Before  giving  a  proof  valid 
for  all  of  the  postulates,  it  will  be  instructive  to  deal  with 
a  selected  few  of  them  singly  by  means  of  simple  figures, 
as  in  the  preceding  lecture.  Postulate  (1)  is  evidently 
satisfied.  In  Fig.  14  the  two  points  A  and  B  determine 
the  pathocircle  a  of  the  (9-bundle. 

Fig.  15  exhibits  the  fact  that  postulate  (8)  is  verified. 
Point  B  is  between  A  and  C  and  between  C  and  A;  neither 
A  nor  C  is  between  the  other  two  of  the  three  points; 
of  course,  no  point  can  move  through  the  absent  0. 


ANOTHER  GEOMETRIC  INTREPRETATION      79 


Let  us  next  have  a  look  at  postulate  (12).  But  we 
must  premise  some 

Definitions. — A  pair  of  points,  A  and  B>  of  a  patho- 
circle  is  a  segment  AB  or  BA;  A  and  B  are  its  ends;  the 
points  between  them  are  the  segment's  points. 


Fig.  16. 

It  is  easy  to  see,  Fig.  16,  that  the  Pasch  postulate  (12) 
is  verified.  A,  B,  C  are  three  points  not  on  a  same 
pathocircle;  they  determine  three  segments,  AB,  BC,  CA; 
the  pathocircle  a  going  through  AB,  one  of  the  three, 
goes  through  another,  BC. 


80  MATHEMATICAL   PHILOSOPHY 

Let  me  suggest  that,  as  an  exercise,  you  make  a  figure 
illustrating  that  the  theorem  (corresponding  to  Hilbert's 
theorem  5)  dealt  with  in  the  preceding  lecture,  is  verified 
in  the  present  interpretation. 

Let  us  turn  to  the  parallel  postulate  (13).  That  it  is 
satisfied  is  clear  in  the  light  of  Fig.  17.  The  given 
pathocircle  is  a;  A  is  a  point  not  on  a;  through  A  there 
is  evidently  one  and  but  one  pathocircle  b  having  no 
point  in  common  with  a;  a  and  b  are,  of  course,  parallel  to 
each  other.     This  postulate,  as  you  know,  is  the  Euclidean 


Fig.  17. 

postulate  par  excellence — the  one  that  mainly  distin- 
guishes Euclidean  geometry  from  the  famous  non-Eucli- 
dean geometries  of  Lobachevski  and  Riemann.  And  so 
you  see,  in  passing,  that  all  interpretations  of  HaF  or 
II AF'  yield  doctrines  of  Euclidean  type — in  the  sense  that 
in  them  the  foregoing  postulate  of  parallels  is  satisfied: 
they  all  of  them  contain  some  theorems  whose  proofs 
depend  upon  that  postulate. 

That  all  of  the  postulates  of  II AF  are  verified  by  the 
meanings  we  have  assigned  to  their  variables  may  be 
quickly  made  evident  by  help  of  the  inversion  transforma- 


ANOTHER  GEOMETRIC   INTERPRETATION      81 

tion,  explained  a  little  while  ago.  Let  us  suppose  our 
field  K  to  be  laid  down  upon  a  Euclidean  plane  ir.  Remem- 
ber that  0  is  absent  from  K  but  that,  below  the  vacant 
position,  ir  has  a  point,  which  we  may  call  0'.  In  K  take 
a  definite  circle  /  for  inversion  circle  having  0  for  center. 
Regard  the  transformation  as  converting  the  points  of 
K  (or  ir)  into  the  points  of  -w  (or  K),  noting  that  0'  of  ir 
and  the  "  ideal  "  point  of  K  are  each  the  other's  trans- 
form; that  the  lines  of  rr  are  converted  into  the  patho- 
circles  of  K,  and  conversely;  and  that,  if,  in  ir,  a  point 
B  is  between  A  and  C  on  a  line,  then  in  K  the  transform 
of  B  is  between  the  transforms  of  A  and  C  on  a  patho- 
circle,  the  transform  of  the  line.  You  see  that  there  is 
thus  established  a  one-to-one  correspondence  between  the 
points  and  lines  of  ir  and  the  points  and  pathocircles  of 
Ky  in  such  a  way  that  all  postulated  relations  among  the 
elements  of  t  hold  equally  among  the  corresponding 
elements  of  K. 

Though  logically  superfluous,  it  will  be  instructive  to 
illustrate  the  matter  a  little  further  by  simple  figures. 
In  Fig.  18,  /  is  the  inversion  circle;  a  is  a  line  in  x;  patho- 
circle  a'  is  the  transform  of  a;  points  A',  B'>  C  are  the 
transforms  of  A>  B,  C;  segments  AB  and  BC  are  con- 
gruent in  the  familiar  sense — in  doctrine  D\\  their 
transforms  A'B'  and  B'C  are  congruent  in  the  new  sense. 
You  see  that  the  postulate  of  Archimedes,  postulate  (20), 
is  verified;  for  as  congruent  segments  stretch  upward  in 
endless  succession  along  a,  their  congruent  transforms 
proceed  on  a'  in  endless  succession  towards  0,  never 
reaching  this  vacant  point-position. 

Fig.  19  illustrates  congruence  of  triangles  in  the  new 
interpretation.  Triangles  ABC  and  A1B1C1  are  con- 
gruent in  7r — in  D\\   their  transforms, — the  new  triangles 


82 


MATHEMATICAL   PHILOSOPHY 


A'B'C  and  A' \B\C \y — are  congruent  in  K — in  the  new 
interpretation. 

Let  us  denote  the  doctrine  arising  from  the  new 
interpretation  of  HAF  by  D3.  Ds  is,  as  you  see,  a  two- 
dimensional  geometry  of  the  points  and  pathocircles  of 
the  field  K  and  is  isomorphic  with  Dx  and  D2.  We  may 
say  that  Di  is  converted,  element  for  element,  figure  for 


B 


A 


Fig.  18. 

figure,  proposition  for  proposition,  into  D3  by  the  inver- 
sion transformation  just  as  D\  is  completely  converted 
into  Z>2  by  the  pole-polar  transformation.  You  thus  begin 
to  glimpse  the  office  and  power  of  what  mathematicians 
call  transformation,  which,  at  the  close  of  the  first  lecture, 
I  named,  as  you  will  remember,  among  the  pillar-concepts 
of  mathematics. 

It  remains  to  give  HaF'  an  interpretation  analogous 


ANOTHER  GEOMETRIC  INTERPRETATION      83 

to  that  we  have  just  given  to  HaF.  I  will  sketch  it 
merely,  inasmuch  as  you  will  find  a  fairly  full  account  of 
it  in  Weber  and  Wellstein's  Elementare  Geometric,  which 
is  the  second  volume  of  their  Encyklopadie  der  Elementar- 
Mathematik — an  excellent  work  handling  in  a  maturely 
critical  way  the  various  elementary  branches  of  mathe- 


FlG.   19. 


matics.  You  should  not  be  misled  by  the  adjective 
Elementare,  for  the  discussions  are  designed  for  advanced 
students. 

By  way  of  a  preliminary,  I  should  say  a  word  respect- 
ing inversion  transformation  of  ordinary  Euclidean  space 
with  respect  to  a  given  sphere.     If  the  radius  be  r  and  the 


84  MATHEMATICAL   PHILOSOPHY 

center  C,  then  two  points,  P  and  P',  on  a  line  through  C, 
are  inverses  of  each  other  provided  the  distance  CP  times 
the  distance  CP' =r2.  You  easily  see  that  to  each  point 
there  corresponds  one  and  but  one  point,  except  that  C 
has  no  correspondent  (in  the  Euclidean  space).  To  annul 
the  exception  we  assume  one  "  ideal  "  point,  or  point  at 
infinity,  to  serve  as  the  transform  of  the  inversion  center 
C.  The  new  space  thus  obtained  is  called  inversion  space. 
The  lines  and  planes  through  C  are  transformed  into  them- 
selves. All  lines  and  planes  not  through  C  are  converted 
respectively  into  circles  and  spheres  through  C. 

And  now  for  the  field  of  our  new  interpretation.  You 
probably  guess  what  it  is  to  be.  Let  S  be  an  inversion 
space;  0  a  chosen  point  in  it.  The  ensemble  of  all  the 
spheres  (including  planes  as  spheres  of  infinite  radius) 
that  go  through  0  may  be  called  the  (9-cluster  of  spheres. 
Now  remove  the  point  0  from  S;  the  cluster  is  now  the 
0-cluster  of  pathospheres;  and  the  cluster  of  circles  bereft 
of  0  will  be  called  the  0-cluster  of  pathocircles.  Our 
field, — let  us  denote  it  by  Kr, — is  composed  of  the  points 
(except  0)  of  S,  the  pathospheres  and  pathocircles  of  the 
O-clusters. 

I  need  hardly  say, — for  you  doubtless  foresee, — that 
our  new  interpretation  of  HaF'  springs  from  agreeing  that 
v\  shall  mean  a  point  of  K';  Vi  shall  mean  a  pathocircle; 
vz  shall  mean  a  pathosphere;  R\  shall  mean  between  in 
the  sense  explained  for  the  field  K;  and  R2  shall  mean 
congruent  in  the  sense  that  the  transforms  of  segments  or 
angles  congruent  in  the  familiar  sense  of  ZV  shall  be 
congruent  in  the  new  sense. 

Call  the  new  doctrine  thus  arising  D3'.  It  evidently 
is  a  three-dimensional  geometry  of  the  points,  patho- 
circles  and   pathospheres   of  the   field   K'   and   matches 


ANOTHER  GEOMETRIC   INTERPRETATION      85 

D\  or  ZV  proposition  for  proposition.  Once  more  let 
me  emphasize  the  fact  that  the  differences — the  very 
striking  differences— of  these  three  geometries  are  psycho- 
logical; logically  the  three  are  one. 

The  next  lecture  will  present  a  won-geometric  inter- 
pretation of  our  two  doctrinal  functions. 


LECTURE  VI 
Non-Geometric  Interpretation 

NOT    ALL    THAT    GLITTERS    IS    GOLD — A    DIAMOND    TEST    OF 

HARMONY TWO-DIMENSIONAL   DOCTRINE    OF    NUMBER 

DYADS    AND    SYSTEMS     THEREOF — THE     THREE-DIMEN- 
SIONAL  ANALOGUE. 

The  interpretations,  or  doctrines,  which  have  hitherto 
concerned  us — D\,  D2,  D3  of  HaF  and  ZV,  ZV,  ZV  of 
HaF' — ought  to  be  called,  and  I  have  called  them, 
geometric  doctrines  because  their  content  or  subject- 
matter, — that  which  the  doctrines  are  doctrines  of  or 
about, — consists  of  things,  whether  sensible  or  purely 
conceptual,  that  are  essentially  and  ultimately  spatial  in 
kind.  The  distinction  is  psychological;  mathematicians, 
not  being  able  to  tell  precisely  what  space  is,  and  dis- 
daining or  affecting  to  disdain  psychology,  may  ignore 
the  distinction,  if  they  like — such  asininity  not  being 
penalized  by  municipal  law  in  any  land.  Let  us  not  be 
so  uncandid  or  so  dull  as  to  ignore  the  essential  distinction 
between  spatial  and  non-spatial  doctrines  merely  because 
they  happen  to  have  the  same  form.  Not  all  that  glitters 
is  gold.  Let  us  not  so  easily  lose  our  common  sense — a 
box  of  table  sugar  is  not  a  box  of  table  salt  even  if  the 
two  boxes  are  identical  in  size  and  form. 

In  the  present  lecture  I  invite  your  attention  to  a  non- 
geometric  interpretation  of  our  doctrinal  functions — to  an 
interpretation,  or  doctrine,  to  be  properly  called  non- 
geometric  because,  though  the  same  in  form  as  the  fore- 

86 


NON-GEOMETRIC  INTERPRETATION  87 

going  geometries,  it  deals  with  non-spatial  things  and  so 
has  a  non-spatial  content.  Some  years  ago  I  asked  Mr. 
Wellington  Koo,  then  a  student  at  Columbia  University 
and  a  pupil  of  mine,  a  brilliant  pupil,  in  analytical  geome- 
try, to  tell  me  what  the  Chinese  word  for  geometry  means 
as  a  word.  He  replied :  "  It  means  show  by  a  figure." 
In  the  interpretation  we  are  about  to  study  we  can  have 
no  figures,  for  figures  are  spatial  affairs.  This  necessity 
of  getting  on  without  figures  is,  in  a  sense,  fortunate — 
fortunate  as  an  intellectual  discipline — for,  in  the  absence 
of  sensuous  representation  by  figures,  we  shall  be  driven 
to  a  kind  of  sheer  thinking.  And  this  warning,  I  hope, 
will  prepare  you  for  the  needed  effort. 

As  in  the  previous  lecture,  I  will  deal  first  with  HaF. 
At  a  later  stage  of  our  course,  the  nature  of  what  is  called 
the  system  of  real  numbers  may  be  discussed.  But  for  the 
purposes  of  the  present  lecture,  I  shall  assume  that  you 
are  sufficiently  acquainted  with  the  system,  merely  remind- 
ing you  that  it  is  composed  of  the  positive  and  negative 
integers;  the  ordinary  fractions;  the  irrationals,  such 
as  V2,  "^7;  and  the  transcendental  numbers,  like  e  and  71-, 
for  example.  By  the  term  number  I  shall  mean  a  real 
number.  In  order  to  indicate  the  nature  and  the  field 
of  our  new  interpretation,  it  will  be  convenient  to  make 
use  of  this  definition:  If  ay  b,  c  be  three  numbers,  b  will  be 
said  to  be  between  a  and  c  (or  c  and  a)  when  and  only  when 
a>b>c  or  a<b<c,  where  > means  greater  than  and< 
means  less  than. 

The  new  field  of  operation — which  may  be  denoted  by 
N — consists  of  all  dyads  (x,  y)  of  real  numbers;  that  is, 
of  all  ordered  pairs  (x,  y),  where  by  ordered  I  mean  that 
(x,  y)  will  not  be  the  same  as  (y,  x)  unless  x=y.  It  is,  of 
course,  understood  that  the  dyads  (#1,  yi)   and   (#2,  V2) 


88  MATHEMATICAL   PHILOSOPHY 

are  distinct  unless  xi  =  X2  and  yi=y2.  You  see  that  the 
field  is  non-spatial,  non-geometric,  for  numbers  and  num- 
ber dyads  have  no  essential  reference  to  space  and  would 
continue  to  be  perfectly  good  objects,  or  subjects,  of 
thought  if  all  spatial  sense  and  all  conception  of  space 
were  to  vanish;  symbols  for  numbers  and  for  dyads  do 
indeed  occupy  room,  but  numbers  themselves  and  dyads 
do  not. 

And  now  it  is  time  to  say  that  our  non-geometric 
interpretation  of  IiAF  arises  from  assigning  to  the  postu- 
late variables  constant  values,  or  meanings,  as  follows: 
v\  will  mean  a  dyad  of  N;  V2  will  mean  a  system  of  dyads, 
i.e.,  the  dyads  satisfying  an  equation  of  the  form  Ax-\- 
By  +  C  =  0,  where  either  A  or  B  is  not  zero,  i.e.,  A  ^0 
or  B^O;  Ri  will  mean  between  in  the  sense  that,  if 
C*i>  yi)>  (x2,  ^2)  and  (X3,  ^3)  are  three  dyads  of  a  same 
system,  (X2,  3^2)  will  be  said  to  be  between  (xi,  yi)  and 
C*3>  y3)  if  and  only  if  X2  is  between  x\  and  #3  or  y2  is 
between  y\  and  yz\  and  R2  will  mean  congruent  in  the 
sense  that  two  dyadic  pairs  (xi,  yi),  (#2,  ^2)  and  (#3,  3>3)> 
(*4,  ^4), — that  is,  two  segments  (xi,  yi,)(x2,  ^2),  (x3,  ys) 
(#4,  V4), — will  be  said  to  be  congruent  when  and  only 
when  V(xi  -x2)2  +  (yi  -y2)2  =  \^(x3  -x4)2+(y2  -y*)2; 
with  a  like  meaning  for  congruence  of  angles  to  be  give 
later. 

Are  the  postulates  in  HAF  verified  by  the  meanings 
thus  assigned?  It  will  be  very  instructive  to  examine 
the  matter  somewhat  carefully. 

Postulate  (1). — Let  Ax-\-By-\-C  —  0  be  an  undeter- 
mined system  s;  d\,  d-2,  any  two  dyads  {x\,  y\),  (X2,  72) 
of  field  N;  d\  and  d2  will  belong  to  s  when  and  only  when 

Axi+Byi+C  =  0, 


1  Ax2+By2  +  C  =  0; 


NON-GEOMETRIC   INTERPRETATION 


89 


three  cases  are  possible  and  only  tnree:  (a)  xi  =  x2,  yi?^; 
(0)  xi9±x2,yi=y2;  (7)  xi^x2,  yi^y2.     In  (a)  5  =  0  and 

C  C 

—  =  -xi=  -x2;    in    (0)  ^=0,    and     j;=-yi=-y2;    in 

(7)  plainly  ^ ^0,  2?^0,  and  if  C  =  0,  then 


A 
B 


X\ 


y2 

y 
X2 


I  yi 

#1  1 

1  y2 

and 

B 
C 

#2  I 

Xi   yi 

Xi   yi 

■*2  y2 

*2  y2 

but  if  C  5^0,  then 

A 
C 


so  that  in  all  cases  s  is  determined,  and  postulate  (1)  is 
verified. 

Postulate  (2). — The  question  is:  do  every  two  dyads  of 
a  system  determine  it?  Let  Ax+By  +  C  =  0  be  a  system 
Sy  (*i>  3>i)  (#2,  ^2),  any  two  dyads,  di,  d2,  of  s.  These 
dyads,  by  postulate  (1),  do  determine  a  system  /,  say 
A'x+B'y  +C"=0.  We  are  to  show  that  s  and  j'  are  the 
same  system. 

If  A=Q,  then  A' =0,  for  ^   is   ixu  -  —  ) 


and  d2  is 


(*2,  ~^j;  nence 

(I) 


BA'xi-B'C+BC'=0, 

BA'x2-B'C+BC'  =  0, 


since  g?i  and  d2  belong  to  /;    as  x\  t±x2,  A'  =  0;    so  s'  is 
i?'y +G"=0;  hence  2?'y  1  +  C  =  0,  and  as  Ji  belongs  to  s, 


90 


MATHEMATICAL   PHILOSOPHY 


c   c 

By\-\-C  =  0;  so  —  =  — ,  and  hence  s'  is  the  same  system  as 
B      B 

s.     If  B  =  0,  the  same  identity  results. 

If  A 9*0  and   5^0,   then,  if  C  =  0,   C'=0,   for  di  is 

/     A  \  *  •  I     A  \    aa    yx     y*  f 

I  x\,  —  —  xi  I,  d2  is  I  x2,  —  -^x2  ),  and  —  = = ;    trom 

\  B     J  \  B     I  B         x\         X2 

the  last  we  see  that  x\9*X2,  y\9*y2\   from  the  equation  of 

s'  we  have 

A'Bxi-AB'xi+BC'^Q, 


(2) 


A'Bx2-AB'x2+BC'=0; 


if  xi  =0  or  #2=0,  then  C"=0,  as  B9*0;  if  #i^0,  #2  5^0, 
divide  (2)  by  x\  and  #2  respectively  and  then  subtract; 
so  it  is  seen  that  C=0.     Hence  s'  is  A'x-\-B'y  =  0,  and, 

as  A'xi+B'yi  =  0,   —  = =  —  and  so,  again,  s'  and  j 

i>  x\     B 

are  the  same. 

Finally,  if  A 9*0,  B^0  and  CV0,  then,  by  the  fore- 
going reasoning,  A' 9*0,  B'  9*0  and  C  9*0.     Hence 


B' 


1  y1 
1  y2 

A 
B 

and 

B' 

a 

X\    I 
X2    I 

x\  y\ 

X2    V2 

xi  yi 
X2  y2 

B 


and  so  in  this,  as  in  all  other  cases,  s  and  s'  are  identical. 
And  postulate  (2)  is  verified. 

Postulate  (7). — This  is  the  next  postulate  in  HaF.  It 
is  satisfied,  for  any  system  Ax-\-By-\-C  =  0  is  evidently 
satisfied  by  infinitely  many  dyads,  and  it  is  evident  that 
no  system  contains  all  the  dyads  of  N. 

Postulate  (8). — If  d\{x\,  yi),  ^2(^2,  ^2),  d^{xz,  y%)  belong 
to  a  same  system  and  if  ^2  is  between  d\  and  d%,  then 
X2  is  between  x\  and  X3,  or  y2  is  between  yi  and  y3;  then, 


NON-GEOMETRIC   INTERPRETATION 


91 


by  definition,  #2  is  between  #3  and  xi,  or  j2  is  between 
y3  and  yi,  and  so  dz  is  between  J3  and  d\.  Hence  the 
postulate  is  satisfied. 

Postulate  (9). — It  is  evident  that  in  determining  a 
dyad  of  any  given  system  we  can  assign  the  x  (or  y)  at 
will.  Now  let  d\{x\y  y\)  and  d^{x^  V3)  be  two  given 
dyads  of  any  given  system  s;  let  ^2(^2,  V2)  be  a  dyad  of 
j-  such  that  *i<#2<#3;  then  d2  is  between  d\  and  d%\ 
next  let  d\{x^  y±)  be  such  that  x\<xz<x±\  then  ^3  is 
between  d\  and  J4.     Hence  the  postulate  is  satisfied. 

Postulate  (10). — We  need  consider  only  four  possi- 
bilities: (a)  A=0,  and  s  is  By  +  C=0;  (0)  5-0,  and  j 
is^*+C  =  0;  (7)  A^0y  B^O,  C=0,  <md  s  is  Ax + By  =0; 
(5)  J^0y  B^O,  C^O,  and  s  is  Ax -\- By +C  =  0. 

You  know  that  of  three  numbers  one  and  only  one  is 
between  the  other  two.     In  (a)  any  three  dyads  of  s  are 

of  the  form  (xi,  —  —J,  lx2,  _^)>   (#3,  ~J>)>    hence  one 

and  only  one  of  the  x's  is  between  the  other  two,  and  so, 
too,  of  the  dyads;  in  (0)  like  reasoning  leads  to  the  same 
conclusion;    in  (7)  let  di(xu  yi),  d2(x2yy2),  ^3(^3,  y 3)  be 


A 


any  three  dyads  of  s;  then  — 

B 


3'3     , 

— ;    hence 

x3 


y_i=  _y_2 

X\  X2 

no  two  x's  (or  y's)  are  equal  for,  if  they  were,  the  corre- 
sponding y's  or  (x's)  would  be  equal  and  we  should  not 
have  three  distinct  dyads;  hence  one  and  only  one  of  the 
x's  (and  also  one  and  only  one  of  the  y's)  is  between  the 
other  two;  hence  so,  too,  the  dyads;  finally,  in  (5)  we 
have 


A 
C 


I  yi 

1  y2 

1  y3 

I  y2 

=  - 

1  y3 

1  yi 
x-s  y-A 

xi   yi 

X2  y-i 

1  x2  y-2 

x-i,   y3 

xi   yi 

92  MATHEMATICAL    PHILOSOPHY 

since  any  two  of  the  dyads  determine  s;  if  two  of  the  y's 
were  equal,  then  A  =  0,  contrary  to  hypothesis,  unless  the 
corresponding  x's  were  also  equal,  but  then  we  should  not 
have  three  distinct  dyads.  Hence  one  and  only  one  of 
the  y's  (or  x's)  is  between  the  other  two,  and,  the  same 
being  consequently  true  of  the  dyads,  the  postulate  is 
verified. 

The  next  postulate  in  HaF  is  the  beautiful  postulate 
(12).     First,  however,  we  must  have  a 

Definition. — A  pair  of  dyads,  d\(x\,  yi)  and  J2  (#2,  V2) 
of  an  s,  is  a  segment  d\d,2  or  ^2^1;  d\  and  d%  are  its  ends; 
all  dyads  between  the  ends  are  the  segment's  dyads. 

Postulate  (12).— Let  us  notice,  in  the  first  place,  that, 
taken  two  at  a  time,  three  dyads,  di(xi,  yi),  ^2(^2,  ^2), 
^3(^3,  V3),  not  belonging  to  a  same  system,  determine  three 
systems,  Si9  S2,  S3,  as  follows: 

x—X2_y—y2_ 
Si  :  =  —  Xi, 

x—x3    y—ys 

x—x3_y—y3_ 

S2  '•  —  —  X2, 

x—xi    y—yi 

x—x\    y—yi 
S3  •  =  —  X3 ; 

x—xo    y—y2 

it  is  plain  that  there  is  but  one  restriction  on  the  X's, 
namely,  Xi^i,  X2?^i,  X3?^i;  for,  except  for  the  inequali- 
ties, the  given  dyads  would  not  be  distinct.  Looking  at 
S\y  for  example,  you  see  that,  when  the  variable  dyad 
d(x,  y)  is  between  J2  and  J3  (i.e.,  when  it  belongs  to  the 
segment  ^2^3),  Xi  is  negative;  and  that,  if  Xi  is  negative 
(and  neither  zero  nor  00 ),  d  is  in  the  segment  ^2^3. 
Clearly  the  same  statement,  mutatis  mutandis,  is  valid  for 
X2  and  X3. 


NON-GEOMETRIC   INTERPRETATION 


93 


Solving  the  foregoing  equations  for  x  and  yy  we  get 

X2  —  Xi^3 


for  si 


for  S2 


x  = 


for  S3 


x  = 


I-Xi 

y2-Xiy3 
I-Xx    ; 

#3  —  X2#l 

I  —  x2 
V3-x2yi 
i-x2  5 

Xi  —  X3#2 
I  -  X3 

yi-X3y2 


1 


x3 


Now  let  us  suppose  that  Ax-\-By-\-C  =  0  is  a  system  j 
not  containing  any  of  the  dyads  di>  do,  d%.  The  condi- 
tions that  s  shall  contain  a  dyad  of  each  of  the  systems 
*i>  S2,  S3,  are  respectively 


Xi  = 


X2 


x3 


Ax2+By2+C 
Axz+Bya+(? 

Axs  +  By3  +  C 
Axi+Byi+C' 

Axi+Byi+C, 

Ax2+By2+C' 


We  have,  as  you  see,  XiX2X3  =  i;  hence  none  of  the  X's  is 
negative  or  else  two  (and  only  two)  of  them  are  negative. 
Now  suppose  that  s  contains  a  d  in  the  segment  d\d2; 
then  X3  is  negative;   hence  Xi  or  \2  is  negative,  and  so  s 


94 


MATHEMATICAL   PHILOSOPHY 


C   B 
C  B' 
A    B 
A'  B' 

>  y-- 

A    C 

A'  a 

A  B 
A'  B' 

contains  a  d  of  segment  d2dz  or  segment  d\d%.     Hence, 
you  see,  our  postulate  is  verified. 

Postulate  (13). — That  this  postulate  of  parallels  is 
verified  in  our  new  interpretation  may  be  quickly  seen  as 
follows.  Let  Ax-\-By-\-C  =  0  be  any  given  system  s,  and 
let  d(x',  y')  be  any  dyad  not  belonging  to  s.  Then  any 
system  s'  containing  d  is  A'{x  —  x')  -\-B'{y  —  yr)  =  0,  or 
A'x+By'+C'=0  where  C'=-UV +£'/).  Solving  s 
and  s'  for  x  and  y,  we  get 


x  —  — 


the  two  terms  of  neither  fraction  can  be  zeroy  for,  if  they 

ABC 
were,   then   —  =  —  =  — ,    and   s   and   s'   would    coincide, 
'  A'    B'     C 

contrary   to   hypothesis;    hence  x   and   y   have   definite 

finite  values  and  accordingly  s  and  s'  have  a  common 

dyad  (x,  y),  except  when  the  denominator  is  zero,  but 

this  can  happen  when  and  only  when  -^}  —  ~^,  and  hence 

B      B 

there  is  one  and  only  one  s'  having  no  dyad  in  common 

with  s,  this  unique  s'  being  parallel  to  s.     And,  as  you 

see,  the  postulate  is  satisfied. 

Before  examining  postulate  (14)  we  require  a 

Definition. — If  d'(x',  yf)  be  a  given  dyad  of  a  system 

s,  any  dyad  d{x>  y)  will  be  said  to  be  on  the  one  side  or  on  the 

opposite  side  of  d'  according  as  x>  x'  or  <x',  except  when  s 

is  of  the  form  Ax+C  =0  and  then  the  distinction  of  sides 

will  depend  on  whether  y>y'  or  y  <y'. 

Postulate  (14). — Let  d\{x\,  y{)  and  d2(x2,  y2)  be  any 

segment  d\d2\    let  d'ix'^  y')   be  any  given  dyad  of  any 


NON-GEOMETRIC  INTERPRETATION  95 

given  system  s;  it  is  clear  that  there  is  in  s  at  least  one 
dyad  d"(x",  y")  such  that  d\d2  is  congruent  with  d'd'\ 
i.e.y  such  that 

V(*i-*2)2  +  (yi-y2)2  =  ^/(x'-x"y+{y'-y")\ 

for  x"  is  at  our  disposal  and  y"  is  a  function  of  it.  But 
is  there  in  s  more  than  one  such  d"1  We  know  that  s  is 
y=mx-\-b  or  else  ;v=m/y+£/;  let  us  use  the  former, 
for  the  reasoning  will  be  the  same  as  for  the  latter.  If 
there  be  a  second  d">  denote  it  by  d'"(x'",  y'"),  where 
x'"  =x" '  +  5;  then  since  did2,  d'd",  d'd'"  are  congruent,  we 

vV-*")2+(y'-/')2  =  •vV-*'"2  +  (y'-)y"')2; 

note  that  y'  =  mx'  +b,  y"  =  mx"  +b,  y'"  =  m{x"  -f-  8)  +b;  sub- 
stituting these  values  in  the  last  radical  equation,  and 
simplifying,  we  get  82+2(x"  —  x')8  =  0;  whence  5=0  or 
5  =  2(x'  —x");  the  former  value  of  8  gives  x"  =x'",  and  so 
does  not  give  a  second  d";  the  latter  value  of  8  gives 
x'"  =x"  —  2{x"  —  x'),  and  so  there  is  one  and  but  one  other 
d"\  now  note  that  x'"  —  x'  =  —(x"—x');  hence  if  one  d" 
is  on  one  side  of  d',  the  other  d"  is  on  the  other  side.  And 
so  the  postulate  is  verified. 

Postulate  (15). — This  postulate  is  so  manifestly  satis- 
fied that  we  need  not  tarry  to  prove  the  fact. 

Postulate  (16). — That  this  postulate  is  verified  may  be 
readily  proved  as  follows:  Let  d\(xi>  yi),  ^2(^2^2)  and 
^3(^3,  ys),  three  dyads  of  any  given  systems  s,  be  such 
that  the  segments  did2  and  ^2^3  have  in  common  no  dyad 
save  ^2;  let  d\,  dz  ■>  ds,  three  dyads  of  any  given  system  /, 
be  such  that  d<i  is  the  only  dyad  common  to  the  segments 
d\d%  and  d^dz  .  Let  d\d2  be  congruent  with  di'd^y 
and  d2^3  with  d^dz  \  we  are  to  prove  that  did-s  and  d\'dz 
are  congruent.     We   may  take  s  to   be  y=mx-\-b,  and  s' 


96  MATHEMATICAL   PHILOSOPHY 

to  be  y—m'x+b';  then,  since  d\d2  and  d\'d2  are  con- 
gruent Vi-\-m2(xi—  x2)  =  y/i+m'2{xi  —  x2);  so,  too, 
Vi+m2(x2  —  xz)  =  Vi  —  m'2(x2  —X3') ;  whence,  by  addi- 
tion, Vi+m2(xi—  xs)  =  Vi  —  m'2(xi—X3');  but  this  last 
equation  tells  us  that  did3  and  d\'dz  are  congruent;  and 
so,  we  see,  the  postulate  is  verified. 

Before  attacking  postulate  (17)  let  us  make  due 
preparation  for  it  in  the  way  of  a  simple  theorem,  some 
definitions  and  a  little  acquaintance  with  a  very  funda- 
mental kind  of  algebraic  transformation. 

Theorem. — Every  system  s  separates  the  remaining 
dyads  of  N  into  two  classes  such  that,  if  d\  and  d2  be  any 
two  dyads  the  segment  d\d2  contains  or  does  not  contain 
a  dyad  of  s  according  as  the  given  dyads  belong,  one  of 
them  to  the  one  class  and  one  of  them  to  the  other,  or  both 
of  the  dyads  belong  to  the  same  class.  [The  theorem  is 
the  correspondent  of  Hilbert's  theorem  5.] 

The  proof  is  not  difficult.  The  given  system  s  is  of  the 
form  (1)  x  =  k  or  of  the  form  (2)  y  =mx-\-b.  If  s  be  of 
form  (1),  it  is  clear  that  the  classes  required  are  respect- 
ively composed  of  dyads  for  which  x>  k  and  of  those  for 
which  x<k.  Next  suppose  s  to  be  y=mx-\-b.  Let 
di(x\,  yi)  and  d2(x2,  y2)  be  any  two  given  dyads  not 
belonging  to  s.  It  is  plain  that  there  is  a  system  si, 
y  =mx-\-b\y  containing  d\,  and  a  system  s2,  y  =mx-\-b2y 
containing  d2;  so  that  yi=mx\+bi  and  y2=mx2+b2. 
The  dyads  d\  and  d2  determine  a  system  sf,  namely, 

x— x\  _y  —  yi 

X—X2    y—y2> 

now  let  d(x,  y)  be  the  dyad  common  to  s  and  s';  then 

x—xi     m{x—x\)-\-b—b\ 
x—x2     m{x—X2)-\-b—b2' 


NON-GEOMETRIC  INTERPRETATION  97 

whence 

x—xi  =  p(b  —b\)  -f-  (j  —  pm), 

x  —X2  =  p{b  —£2)  -s-  (j  —pm), 

where  p  is  a  proportionality  factor.  You  see  that  x—xi 
and  X—X2  are  unlike  or  like  in  sign  according  as  b—  b\ 
and  b—  b%  are  unlike  or  like  in  sign;  that  is,  d  is  between 
or  not  between  d\  and  di  according  as  b  is  between  or  not 
between  b\  and  #2;  hence  the  theorem.  We  may  agree  to 
say  that  the  dyad  d\{x\,  y\)  is  on  the  positive  or  the 
negative  side  of  s  according  as  b\>  or  <b. 

Definitions. — If  d  be  a  dyad  of  a  system  s,  the  dyads 
of  j  on  a  same  side  of  d  constitute  a  half-system  emanating 
from  d.  [So  it  is  seen  that  any  dyad  of  an  s  separates  the 
remaining  dyads  of  s  into  two  opposite  half-systems.] 
A  pair  of  half-systems,  h  and  k,  emanating  from  a  dyad  d 
and  not  belonging  to  a  same  system,  is  an  angle  (h,  k); 
d  is  the  angle's  vertex,  and  h  and  k  its  sides;  its  interior  is 
the  class  of  dyads  such  that,  if  d\  and  ^2  be  any  two  of  them, 
the  segment  d\d<i  contains  no  dyad  of  h  or  k;  its  exterior 
consists  of  all  other  dyads  of  N  except  d  and  the  dyads 
of  h  and  k.  Let  d  and  d'  be  the  vertices  of  two  angles 
(h,  k)  and  (hr,  k') ;  let  dd\  and  dd'2  be  two  segments  of 
h  and  k  respectively,  and  let  d'd\  and  d'd^'  be  segments  of 
h'  and  k'  respectively;  suppose  dd\  is  congruent  with 
d'd\  and  ddi  with  d'd2r\  then,  if  d\d2  is  congruent  with 
d\'d2  ■>  we  shall  say  that  the  given  angles  are  congruent. 
[Note  that  we  have  defined  congruence  of  angles  in  terms 
of  congruence  of  segments.  Note  also  and  note  carefully 
that,  though  we  have  for  the  sake  of  convenience  used 
such  terms  as  angle,  vertex,  side,  and  so  on,  which  smell 
of  geometry  and  suggest  space,  there  is  in  such  use  no 
logically  implicit  geometric  or  spatial  reference  whatever. 


98  MATHEMATICAL   PHILOSOPHY 

The  use  of  those  terms  here  is  purely  metaphorical  and, 
had  we  desired  to  dispense  with  their  use,  it  would  not 
have  been  difficult,  as  you  no  doubt  see,  to  do  so.] 

Let  me  now  explain  briefly  a  simple  but  exceedingly 
important  algebraic  transformation  which  will  be  very 
helpful  in  dealing  with  postulate  (17).  Consider  the  pair 
of  equations 

[  x—x'  cos  d—  y'  sin  d+a, 
^'     \  y  =x'  sin  6+y'  cos  6+b; 

solving  these  for  x'  and  y',  we  get  the  pair 

,      J  x'  =  x  cos  6  +3;  sin  0  —  a  cos  6  —  b  sin  0, 
^        I  y'  =  —  x  sin  0+y  cos  0  —  b  cos  6-\-a  sin  0; 

you  notice  that  (t)  and  (t')  are  equivalent,  either  pair 
being  obtainable  from  the  other.  Either  pair,  say  (t) 
defines  a  dyad-to-dyad  transformation;  that  is,  given  a 
dyad  (x'y  y'),  there  corresponds  to  it,  by  virtue  of  (t) 
a  definite  dyad  (x,  y),  and  conversely.  Of  two  dyads 
thus  related,  we  say  that  each  is  the  other's  transform 
or  that  each  is  converted  or  transformed  into  the  other. 
I  wish  to  call  your  attention  to  four  further  properties 
of  the  transformation.  One  of  them  is  that  the  dyads 
of  a  system  are  converted  into  the  dyads  of  a  system.  To 
prove  this  proposition,  take  any  system  Ax+By+C 
=  0,  replace  x  and  y  by  their  values  from  (t),  simplify, 
and  note  that  you  then  have  the  equation  of  a  system. 
Thus  a  dyad-to-dyad  transformation  is  also  a  system-to- 
system  transformation:  the  transform  of  a  system  is  a 
system.  Another  property  of  the  transformation,  show- 
ing its  power,  is  that,  owing  to  the  presence  of  three 
undetermined  quantities,  or  parameters  as  they  are  called 
— a,    b    and    6 — ,    we    can    convert    any    given    system 


NON-GEOMETRIC  INTERPRETATION  99 

(i)  Ax-\-By+C  =  0  into  any  given  system  (2)  A'x+B'y  + 

C  =0  and,   at  the  same  time,   any  given  dyad  of  the 

former  into  any  given  dyad  of  the  latter;    to  show  this 

possibility,  transform  (1)  as  above  indicated,  then  equate 

the  two  ratios  of  the  coefficients  in  the  resulting  equation 

to  the  corresponding  ratios  taken  from  (2);    these  two 

equations    (two  conditions   on   a,   b   and    0)    insure   that 

(1)  has  (2)  for  its  transform;    but  our  three  parameters 

can    satisfy   a   third    condition;     notice   what   it   is;     let 

d\{xiyyi)  be  the  given  dyad  of  (1),  and  d\'{xi\  yi')  the 

AC  A'         C 

given  dyad  of  (2) ;  theny 1  =  -  — xl  -  -  and  y  /  =  -~xi'  -  — ; 

d\  is  to  be  converted  into  d\   and  this  gives  the  third  con- 

I  A'  C'\ 

dition,  which  is  that  X\  =x\    cos  Q~\-[—pc\-\-  —  J  sin    6-\-a 

or  an  equivalent  one  obtained  from  the  second  equation 
of  (t).  The  writing  out  of  the  three  conditions  and 
solving  them  for  a,  b  and  0  involves  a  little  finger  work 
but  no  logical  difficulty.  You  may  wish  to  perform  the 
task  as  an  exercise.  Again,  any  one  of  our  dyad-to-dyad 
transformations  converts  any  given  segment  into  a 
congruent  segment.  I  say  "  any  one  of  our  dyad-to-dyad 
transformations,"  for  we  have  many,  infinitely  many, 
of  them,  depending  on  the  values  we  assign  to  the  par- 
ameters a,  b  and  0.  To  prove  the  property  in  question 
let  the  segment  be  determined  by  di(xi,  yi)  and  ^2(^2,  y2); 
in  V  (xi  —  *2)2  +  (yi  —  y2)2  replace  xi,  yiy  X2,  y-2  by  their 
transforms  xi  cos  0—  yi  sin  d+a,  X\  sin  0+yi  cos  6+by 
x2  cos  6—  y-z  sin  0+<z,  X2  sin  0+y2  cos  0+&,  simplify  and 
then  note  that  the  radical  expression  has  suffered  no 
change.  Finally,  any  one  of  our  transformations  leaves 
the  order  of  dyads  unchanged;  that  is,  if  di,  ^2  and  d\>,  are 
converted   respectively  into  di,  d~2    and  ds,  then,   if  d2 


100  MATHEMATICAL   PHILOSOPHY 

be  between  d\  and  dz,  d<i  will  be  between  d\  and  d%.  Let 
dif  d2,  dz  belong  to  s,  y=mx+c;  then  yi=mxi-\-c,  y2  = 
mx2-\-c  and  yz=mxz-\-c;   then  from  (t)  we  have 

x\  =(cos  6-\-m  sin  d)x\-\-(c  —  b)  sin  6  —  a  cos  0, 
X2  =  (cos  0  +  m  sin  0)#2  +  (c  —  b)  sin  0  —  a  cos  0, 
#3'  =  (cos  0+w  sin  0)xz  +  {c—  b)  sin  0—  a  cos  0; 
hence 

*i'  —xr2  =  (cos  0  +  m  sin  0)  (#1  —#2), 
X2  —xf3  =  (cos  0+?n  sin  0)(#2— #3); 

hence  if  #i>#2>#3  or  xi<x2<X3,  then  #i'>a;2'>.*3'  or 
xi  <X2  <X3\  that  is,  if  ^2  is  between  Ji  and  dz,  ^2'  is 
between  d\  and  ^3'. 

From  the  invariance  of  congruence  and  of  order,  or 
betweenness,  it  follows  that,  if  the  angle  (hf,  k')  be  the 
transform  of  the  angle  (A,  k),  the  interior  of  the  former 
is  the  transform  of  the  latter's  interior  and  that  the 
angles  are  congruent. 

With  the  foregoing  equipment  we  may  proceed  to  the 
examination  of 

Postulate  (17). — Let  me  ask  you  to  read  the  postulate 
very  attentively.  It  requires  us  to  prove  the  following 
proposition:  Given  an  angle  (h,  k),  a  system  sy  a  dyad  d  of  s 
and  a  half -system  h'  emanating  from  d,  there  is  one  and  but 
one  half -system  k'  {emanating  from  d)  such  that  the  angle 
(h'y  k')  is  congruent  with  the  angle  (hy  k)  a?id  that  the 
interior  of  (A,  k)  is  on  a  given  side  of  s. 

In  virtue  of  our  dyad-to-dyad  transformation  it  is 
evident  that,  without  loss  of  generality,  we  may  take  the 
sides  h  and  k  to  be  half-systems  belonging  respectively 
to  the  systems  S\,  y  =m\x  and  J2,  y  =m2X-\-b,  and  emanat- 
ing from  their  common  dyad,  say,  d'(xf,  y');  that  we  may 
take  J  to   be  y  =0,  d  to  be  the  dyad  (0,  0)  of  s,    and   h' 


NON-GEOMETRIC   INTERPRETATION  101 

to  be  a  half-system  belonging  to  s  and  emanating  from 
(0,  0).  Let  us  now  choose,  as  we  evidently  may  choose, 
di(xi,  yi)  on  si,  ^2(^2,  y2)  on  S2,  ^3(^3,  0)  on  s  and  ^4(^4,  ji) 
on  y=mx  so  that  the  segments  d'di,  d'do,  shall  be  con- 
gruent respectively  to  ddz  and  dd^;  note  that  dd±  is  part 
of  the  side  k'  of  the  angle  whose  existence  is  to  be  estab- 
lished. We  have  to  show  that  m  may  be  so  chosen  that 
d\d2  and  dsd*  shall  be  congruent.  By  virtue  of  the  given 
congruences,  the  condition  that  did2  and  ^3^4  shall  be 
congruent  and  that  consequently  the  angles  (A,  k)  and 
{h'>  k')  shall  be  congruent  is  readily  found  to  be 


m 


2  — 


(mi  —m.2)2 
(1  -f  W1W2)2' 


there  are,  you  see,  two  real  values  of  m,  of  opposite  signs, 
corresponding  to  the  two  sides  of  j"  (or  h') ;  and  the  postu- 
late is,  accordingly,  satisfied. 

Postulate  (18). — So  plainly  satisfied  as  not  to  detain  us. 

Postulate  (19). — That  this  one  is  satisfied  follows  at 
once  from  our  definition  of  congruence  of  angles  and  the 
fact  that  postulate  (17)  is  satisfied. 

We  now  come,  finally,  to  the  Archimedean  postulate 
of  continuity. 

Postulate  (20).— By  reason  of  the  properties  of  our  dyad- 
to-dyad  transformation  we  may,  without  loss  of  general- 
ity, choose  the  system,  y=0,  for  system  s,  and  for  given 
dyads  of  s  the  dyads  d(0,  0)  and  d'{x',  0).  Let  d\(x\,  0), 
^2(^2,  0),  ^3(^3,  0),  .  .  .  be  such  that  d\  is  between  d  and 
d'  and  between  d  and  J2,  that  d2  is  between  d\  and  dz  .  .  ., 
and  that  the  segments  ddi,  d\d2,  dods,  .  .  .  are  mutually 
congruent.  We  are  to  prove  that  in  the  dyad  series  there 
is  a  dyad  dn(xn,  0)  such  that  d'  is  between  d  and  dn. 
The  series  of  x's  is  an  increasing  or  decreasing  series,  say 


102  MATHEMATICAL   PHILOSOPHY 

increasing.  Then  #1  =#2— #1  =#3— #2  =  .  .  .  =ixn—xn_1. 
The  sum  nx\=xn\  we  choose  n  so  that  nx\>x\  then 
xn>x',  but  x'>0;  hence  x'  is  between  0  and  xn  and  hence 
d'  is  between  d  and  Ja;  which  was  to  be  proved. 

To  prove  the  compatibility  of  postulates  we  have  to 
find  a  set  of  things  regarding  which  the  ^jstulates  make 
true  statements  when  the  things  are  put  in  place  of  the 
variables.  The  better  the  things  are  known,  the  better 
is  the  test.  Now  number  dyads  and  systems  thereof 
are  the  best  known  of  things;  and  so,  in  showing  that 
they  verify  the  Hilbert  postulates,  we  have  established 
their  compatability  by  the  diamond  test. 

Let  us  denote  the  doctrine  arising  from  the  foregoing 
interpretation  of  HaF  by  D±.  D4,  as  I  have  said  and  as 
you  must  now  plainly  see,  is  in  all  strictness  non-geometric, 
having  no  spatial  content.  It  is  purely  algebraic  or 
numerical — a  two-dimensional  theory  of  dyads  and  sys- 
tems of  dyads  of  real  numbers.  In  point  of  form  it  is 
Euclidean,  having  the  same  form  as  Euclidean  plane 
geometry;  but  to  say  that  is  to  say  that  Euclidean 
geometry  has  the  same  form  as  the  Dyad  doctrine.  If 
the  latter  had  happened,  as  it  might  have  happened,  to 
be  developed  prior  to  Euclidean  geometry  and  received 
a  name  proper  to  it,  there  would  be  precisely  as  much 
sense  and  propriety  in  calling  Euclidean  geometry  by 
that  name  as  there  now  is  in  calling  the  Dyad  doctrine 
Euclidean  geometry. 

This  lecture  has  grown,  I  fear,  to  a  wearisome  length. 
Yet  I  must  ask  your  permission  to  continue  long  enough 
to  indicate  very  briefly  an  interpretation  of  HaF'  analo- 
gous to  the  foregoing  interpretation  of  HaF.  The  field 
N'  of  the  interpretation  in  question  is  composed  of  all  the 
triads  (x,  y,  z)  of  the  real  numbers.     The  interpretation 


NON-GEOMETRIC   INTERPRETATION  103 

arises  from  assigning  to  the  variables  in  HaF'  the  following 
values,  or  meanings:  v\  is  to  denote  a  triad  of  N';  vo,  the 
system  of  triads  common  to  a  pair  of  equations,  Ax-\-  By  -f 
Cz+D=0,  Jfx+B'y+C'z+D  =  0;v3,  the  system  of  triads 
satisfying  one  such  equation;  v±  is  to  mean  N'\  Ri  is  to 
mean  between  in  the  sense  that  if  the  triads,  ti(xi,  yi,  Zi), 
*2(*2,  ^2,  z2),  ^3(^3,  ^3,  Z3),  belong  to  a  system  of  the 
former  kind,  then  t%  will  be  between  t\  and  h  when  and 
only  when  #2  is  between  #1  and  #3  or  yz  is  between  y\  and 
^3  or  Z2  is  between  zi  and  Z3;  and  Ro  is  to  mean  congruent, 
or  equal,  in  the  sense  that  the  segment  M2  will  be  con- 
gruent to  segment  hh  when  and  only  when, 


V(*i  -^2)2  +  (yi  -y2)2  +  (zi  -z2)2  = 

V(^3-^4)2  +  (y3-y4)2  +  (z3-Z4)2, 

and,  for  angles,  in  the  sense  analogous  to  that  given  in 
the  preceding  interpretation. 

The  interpretation  is  worked  out  with  some  detail  in 
a  very  interesting  and  enlightening  way  in  the  Elementare 
Geometrie  cited  in  the  preceding  lecture.  The  doctrine, 
ZV>  arising  thus  from  IlAF'  is,  as  you  see,  non-spatial  and 
non-geometric;  it  is  a  purely  algebraic  three-dimensional 
theory  of  triads  and  systems  of  triads  of  real  numbers  and 
is,  of  course,  isomorphic  with  ordinary  Euclidean  geome- 
try of  space. 


LECTURE  VII 
Essential  Discriminations 

DISTINCTION  OF  DOCTRINE  AND  METHOD ANALYTIC  GEOM- 
ETRY AND  GEOMETRIC  ANALYSIS THE  TWAIN  BEGOT- 
TEN   OF    CONVERSE    TRANSFORMATIONS AN    INFINITE 

FAMILY     OF      SISTERS ALL       HERITORS       OF       THEIR 

MOTHER'S     FORM THEIR     COMMON     CHARACTER     AND 

INDIVIDUALITIES — EXCESSIVE     MEANING     OF   CONTENT 

GENERIC    AND    SPECIFIC     MEANINGS    OF    EUCLIDEAN 

AND     NON-EUCLIDEAN THREE     PROPERTIES     COMMON 

TO    POSTULATE    SYSTEMS FERTILITY    AND     COMPEND- 

ENCE  AND  COMPATIBILITY. 

In  this  lecture  and  the  next  one,  I  invite  you  to  join 
me  in  considering  a  variety  of  kindred  matters  closely 
connected  with  the  preceding  lectures.  Some  of  these 
matters  are  suggested  in  the  foregoing  lengthy  title. 
In  course  of  the  discussion  some  of  the  unanswered  ques- 
tions you  have  asked  and  some  others  that  you  are  no 
doubt  prepared  to  ask  will,  I  hope,  receive  suitable 
answers.  I  say  "  some  "  of  them,  for  I  trust  we  are  not 
so  stupid  as  to  be  able  to  answer  all  the  questions  we  are 
able  to  ask.  Let  us  begin  with  one  of  the  questions  that 
must  be  asked  and  can  be  answered  satisfactorily. 

A  Word  about  Analytic  Geometry  and  Geometric 
Analysis. — In  Lecture  V,  I  gave  a  little  introduction  to 
what  is  called  the  analytic  or  algebraic  geometry  of  the 

104 


ESSENTIAL  DISCRIMINATIONS  105 

Euclidean  plane.  We  saw  that,  a  pair  of  axes  and  a 
distance  unit  being  chosen,  to  any  point  P  of  the  plane 
there  belongs  a  pair  (x,  y)  of  real  numbers,  and  con- 
versely; and  that  to  each  line  there  belongs  an  equation 
Ax-\-By-\-C  =  0,  and  conversely.  Now  such  a  pair  and 
such  an  equation  are  respectively  what  we  called  in 
Lecture  VI  a  dyad  and  a  system  of  dyads.  The  question 
is:  Is  not  the  dyad  doctrine  D4  simply  ordinary  Euclidean 
geometry  Di  in  disguise?  I  might  answer,  quite  justly, 
that  D±  is  no  more  and  no  less  Di  in  disguise  than  Di  is 
Z>4  in  disguise.  You  may  now  wish  to  say:  very  well, 
are  not  Di  and  D4  identical?  The  answer  is  no,  for  Di 
is  a  doctrine  about  spatial  things — points  and  lines — 
while  D\  is  a  doctrine  about  non-spatial  things— dyads 
and  systems  of  dyads  of  pure  real  numbers.  Perhaps 
you  would  rejoin,  saying:  Is  not  D±  simply  the  analytic, 
or  algebraic,  geometry  of  the  Euclidean  plane?  It  is 
evidently  just  to  answer:  D±  is  that,  no  more  and  no  less 
than  D\  is  the  geometric  algebra  of  N,  which  is  the  field  of 
number  dyads  and  systems  thereof  just  as  the  plane  is  the 
field  of  points  and  lines.  And  you  know  that  D\ — the 
ordinary  plane  geometry  of  Euclid — is  not  an  algebra. 
The  fact  is  that,  unless  we  are  content  to  confound  things 
that  are  essentially  different,  we  must  here  distinguish 
four  different  things:  namely,  Di,  Di,  and  two  converse 
aspects  of  what  is  in  usage  somewhat  indiscriminately 
called  analytic,  or  algebraic,  geometry  of  the  Euclidean 
plane.  One  of  these  aspects  ought  to  be  called  analytic, 
or  algebraic,  geometry;  and  the  other,  geometric  analysis 
or  geometric  algebra.  "  Ought,"  I  mean,  for  the  sake  of 
philosophic  clarity,  not  necessarily  in  common  every-day 
parlance  or  practice.  Let  us  be  quite  clear  in  this  busi- 
ness.    What  is  commonly  called  the  analytic,  or  algebraic, 


106  MATHEMATICAL   PHILOSOPHY 

geometry  of  the  Euclidean  plane  has  its  birth  in  a  certain 
transformation — a  point-to-dyad  transformation — which 
consists  in  the  fact  that  a  one-to-one  correspondence  sub- 
sists between  the  points  of  the  plane  and  the  number  dyads 
(x,  y)  of  N.  By  virtue  of  this  transformation,  to  any 
given  relation  among  points  in  doctrine  Di  there  corre- 
sponds a  definite  relation  among  dyads  in  doctrine  At; 
and  conversely,  for  the  correspondence  runs  both  ways. 
Do  not  fail  to  note  now  very  carefully,  for  this  is  the  crux 
of  the  matter,  that,  owing  to  the  mentioned  correspond- 
ence, we  can  translate  a  problem  respecting  points  into  a 
problem  respecting  dyads,  then  solve  the  latter  (alge- 
braically) and  finally  translate  the  result  in  terms  of  points, 
thus  getting  a  proposition  in  D\\  and,  conversely,  we  can 
translate  a  problem  respecting  dyads  into  a  problem 
respecting  points,  then  solve  the  latter  (geometrically) 
and  finally  translate  the  result  in  terms  of  dyads,  thus 
getting  a  proposition  in  A:  in  other  words,  we  can  inves- 
tigate algebraically  the  point  relations  making  up  D\ 
and,  conversely,  we  can  investigate  geometrically  the 
dyad  relations  making  up  A-  It  is  now  obvious  that, 
instead  of  calling  both  of  these  converse  procedures 
analytic,  or  algebraic,  geometry,  the  former  ought  to  be 
called  analytic,  or  algebraic,  geometry;  and  the  latter 
geometric  analysis  or  geometric  algebra.  Observe  that 
neither  of  them  yields  a  new  doctrine;  each  of  them  is 
simply  a  new  method  of  establishing  an  old  doctrine; 
and  the  fundamental  distinction  between  the  two  doc- 
trines, A  and  A,  remains  in  undisturbed  serenity. 

You  perceive  at  once  that  the  foregoing  discussion 
applies,  mutatis  mutandis,  to  A'  and  At'. 

The  Possibility  of  Yet  Other  Interpretations  of  HaF  and 
HaF'  . — To  each  of  these  doctrinal  functions  have  now 


ESSENTIAL  DISCRIMINATIONS  107 

been  given  three  geometric  interpretations  but  only  one 
non-geometric  one,  and  the  latter  is  algebraic.  It  is 
natural  to  ask:  are  there  other  algebraic  interpretations? 
The  answer  is,  there  are.  I  shall  not  tarry  to  present 
them,  for  we  have  many  other  things  to  consider,  but  we 
may  pause  a  moment  to  convince  ourselves  of  their 
existence.  Let  us  recall  our  third  interpretation  of 
HaF,  for  example,  giving  rise  to  the  geometric  doctrine 
Dz.  It  is  plain  that  in  it  we  may  replace  point  by  dyad 
and  pathocircle  by  an  equation  determining  a  perfectly 
corresponding  system  of  dyads,  and  thus  obtain  a  new 
algebraic  interpretation  of  HaF  and  therewith  a  new  two- 
dimensional  theory  of  dyads  and  dyad  systems.  And 
so  on — an  algebraic  interpretation  for  each  geometric  one 
and  conversely. 

How  many  geometric  and  how  many  algebraic  inter- 
pretations of  HaF  or  of  HaF'  are  possible  ?  Is  the  number 
finite  or  infinite?  I  will  state— without  giving  the  proof 
— that  each  of  the  two  functions  admits  of  an  infinitude 
of  interpretations  of  either  sort.  And  I  may  add, — again 
omitting  the  proof,  which  is  easy, — that  from  any  given 
interpretation,  whether  geometric  or  algebraic,  one  can 
derive  an  endless  series  of  different  interpretations, 
correspondingly  geometric  or  algebraic,  drawing  them, 
each  out  of  its  predecessor,  unceasingly  as  the  successive 
joints  of  an  infinitely-many-jointed  telescope.  Most  of 
the  interpretations  thus  obtainable  and  the  corresponding 
doctrines  are  devoid  of  interest  for  us  human  beings,  but 
that  statement  is  a  commentary  upon  our  supersimian 
curiosity  and  not  upon  the  intrinsic  merits  of  the  doctrines. 

Do  HaF  and  II AF'  admit  of  interpretations  that  are 
both  non-geometric  and  non-algebraic?  Yes:  each  of  the 
functions  admits  of  an  infinite  variety  of  such  interpreta- 


108  MATHEMATICAL    THILOSOPHY 

tions.  It  is  very  easy  to  prove  it.  To  do  so,  consider 
any  one  of  the  interpretations  we  have  encountered,  say 
the  fourth  one  of  HaF — the  doctrine  At.  Any  other  one 
would  do  as  well.  You  know  that  we  can,  if  we  choose, 
associate  in  our  thought  any  two  given  objects,  0\  and  O2, 
thus  obtaining  a  third  object,  O3  (which  is  simply  Oi 
and  Oo  associated  together).  Now  let  0  denote  some  given 
object  of  thought,  say  the  center  of  gravity  of  the  Milky 
Way  or  Caesar's  love  for  Cleopatra  or  the  taste  of  good 
whiskey — any  specific  thing,  no  matter  what.  Now 
associate  0  with  each  dyad  involved  in  D4;  association  of 
0  with  (xi,  yi)  gives  a  new  object  0\\  association  of 
0  with  (xo,  ^2)  gives  us  another  new  object  O2';  and  so  on. 
Never  mind  how  arbitrary  or  artificial  or  uninteresting  the 
new  objects  may  be,  for  that  is  of  no  logical  importance  at 
all.  Observe  that  we  have  a  one-to-one  correspondence 
between  the  dyads  in  field  N  and  the  objects  0\  which 
we  may  think  of  as  constituting  a  field  M.  You  see  that 
to  a  given  system  of  dyads  there  now  corresponds  a 
definite  class  of  the  0"s,  which  class  we  may,  if  we  like, 
call  a  system  of  0"s.  Let  us  next  agree, — evidently  we 
may  agree, — to  say  that  two  or  more  0"s  satisfy  a  relation 
when  and  only  when  the  corresponding  dyads  satisfy  the 
relation.  You  see  immediately  that,  in  virtue  of  our 
agreement,  or  convention,  the  0"s  of  M  and  the  0' -systems 
verify  the  postulates  of  H&F  just  as  well  as  do  the  dyads 
and  dyad  systems  of  N,  that  is,  perfectly.  And  you  see 
that  there  thus  arises  a  new  interpretation  of  HAF  and  a 
new  doctrine  whose  content  differs  from  that  of  D4  as 
an  0'  differs  from  a  dyad.  If  you  choose  a  difFerent  0 
you  obtain  a  new  kind  of  object  0'  and  a  new  doctrine. 
You  thus  get  as  many  doctrines  as  there  are  objects  0  to 
use.     If  God  has  not  made  an  infinite  number  of  (9's  for 


ESSENTIAL  DISCRIMINATIONS  109 

you,  you  doubtless  see  that  you  can  make  them  yourselves. 
I  grant  that  the  vast  majority  of  doctrines  that  are  con- 
structive in  the  way  indicated  are  trivial — mere  weeds 
of  the  doctrinal  garden;  it  was,  however,  not  our  task  to 
estimate  their  worth,  but  to  demonstrate  their  infinite 
multiplicity. 

Sense  in  which  All  Doctrines  Derivable  from  HAF  and 
HaF'  Are  Like  in  Form,  or  Structure. — Let  me  request 
you  to  remind  yourselves  vividly  of  the  fact  that  each 
of  the  doctrinal  Functions  consists  of  a  system  of  prepo- 
sitional functions,  called  postulates,  and  a  set  of  preposi- 
tional functions  logically  deducible  from  the  postulates 
and  called  theorems.  Be  good  enough  to  recall  also  the 
fact  that,  if  we  replace  the  variables  in  the  postulates 
of  one  of  the  doctrinal  functions  by  admissible  constants — 
a  term  already  explained — we  thereby  obtain  a  doctrine, 
which  is  true,  and  then  called  a  value  of  the  function, 
or  false,  according  as  the  substituted  constants  verify 
or  do  not  verify  all  of  the  postulates.  Because  the  doc- 
trine, whether  true  or  false,  matches  the  doctrinal  func- 
tion, statement  for  statement,  and  because  the  statements 
(propositions)  composing  the  doctrine  and  the  corre- 
sponding statements  (prepositional  functions)  composing 
the  doctrinal  function  are  identical  in  respect  of  form,  we 
say  that  the  doctrine  and  the  function  are  themselves  like 
in  form,  or  structure.  You  see  that,  therefore,  the  infini- 
tude of  values  of  either  one  of  our  doctrinal  functions  and 
the  infinitude  of  false  doctrines  derivable  from  it  are  all 
of  them  like  in  form,  or  structure,  for  each  of  them  is  like 
in  form,  or  structure,  to  the  function  from  which  all  of 
them  are  derivable. 

Senses  in  which  All  Doctrines  Derivable  from  II AF  and 
II AF'  Are  Like  and  Unlike  in  Content,  or  Subject-matter. — 


110  MATHEMATICAL   PHILOSOPHY 

The  doctrinal  functions,  we  have  seen,  have  no  specific 
content,  no  definite  subject-matter,  and  are  neither  true 
nor  false.  On  the  other  hand,  each  of  the  derivable 
doctrines  has  specific  content,  or  subject-matter,  and  is 
true  or  else  false.  Being,  as  we  have  seen,  like  in  form,  or 
structure,  the  doctrines  are  discriminated  among  them- 
selves solely  by  differences  of  content,  or  subject-matter. 
Now,  their  contents,  or  subject-matters,  difFer  in  respect 
of  what  we  may  call  their  meanings.  Query:  Is  there  any 
respect  in  which  the  contents  of  the  various  doctrines  are 
identical?  The  answer  is  that  the  contents  of  all  of  the 
true  doctrines, — of  all  of  the  values  of  the  doctrinal  func- 
tion concerned, — are  identical  in  the  respect  that  the 
various  contents  equally  verify,  or  satisfy,  the  postulates 
of  the  function;  but  such  partial  identity  of  content  can 
not  be  affirmed  of  two  of  the  false  doctrines.  Let  us  now 
confine  our  attention  to  the  true  doctrines  for  it  is  these 
that  we  value.  It  is  perfectly  clear  that  the  meaning  of 
the  content,  or  subject-matter,  of  such  a  doctrine, — the 
meaning,  that  is,  of  the  things  which  the  doctrine  is  a 
doctrine  of  or  about,— -is  not  exhausted  by  the  requirement 
that  the  things  shall  be  verifiers  of  the  postulates.  Ordi- 
nary points  and  lines,  for  example,  or  number  dyads  and 
dyad  systems,  have  countless  uses  and  significances  over 
and  above  the  service  they  render  by  satisfying  the 
postulates  of  HAF.  The  meaning  that  the  content  of  a 
true  doctrine  has  beyond  that  it  must  have  to  verify  the 
postulates  of  the  function  of  which  the  doctrine  is  a  value 
may  be  called  the  content's,  or  subject-matter's,  excessive 
meaning.  Thus  you  see  that  the  infinitely  many  diverse 
doctrines  having  a  given  doctrinal  function  for  their 
common  matrix  are  discriminated  from  each  other  by 
diversities  in  the  excessive  meanings  of  their  contents. 


ESSENTIAL  DISCRIMINATIONS  111 

and  each  doctrine  is  identified  by  something  peculiar  in 
the  excessive  meaning  of  its  content.  The  statement  just 
made  holds  good  for  true  doctrines  only,  for  it  is  evident 
that  to  false  doctrines  the  notion  of  excessive  meaning 
does  not  apply.  How  are  the  false  doctrines  having  a 
common  functional  matrix  discriminated  and  identified? 
I  leave  the  question  to  such  of  you  as  may  be  interested 
to  consider  it. 

We  have  seen  that  all  doctrines,  whether  true  or  false, 
that  have  a  same  doctrinal  function  for  matrix  are  like  in 
form — they  have,  that  is,  the  same  logical  frame  or 
structure.  It  is  pretty  obvious  and  very  noteworthy 
that,  on  the  other  hand,  what  we  have  called  the  excessive 
meaning  of  a  true  doctrine's  content  is  thus  not  logical, 
but  is  purely  psychological.  A  point  and  a  number  dyad, 
for  example,  or  a  line  and  a  pathocircle,  or  a  plane  and  p. 
pathosphere,  though  they  fit  into  the  same  logical  scheme, 
performing  the  same  (logical)  office  in  relation  to  the 
postulates,  are  discriminated  not  only  by  their  differences 
as  concepts, — which  are  psychological  phenomena — but 
also  and  especially  by  the  exceedingly  different  imageries 
or  intuitions  with  which  they  and  the  doctrines  they 
figure  in  crowd  the  mind.  You  are  students  of  philosophy. 
As  such  you  ought  to  be  interested  in  psychology  and  I 
trust  none  of  you  is  afflicted  with  psychological  blindness. 
Not  long  ago,  a  professional  philosopher  in  good  stand- 
ing, told  me  that  he  saw  no  "  psychological  "  difference 
between  a  point  and  a  straight  line.  I  cannot  under- 
stand how  any  student  with  a  feeling  for  psychology  can 
fail  to  have  a  little  quickening  of  pulse  when  he  sees 
clearly  for  the  first  time  the  fact  now  staring  us  in  the 
face:  namely,  that  we  are  living  in  a  world  where  it  is 
possible  to  have  an  infinitude  of  true  doctrines  and  an 


112  MATHEMATICAL   PHILOSOPHY 

infinitude  of  false  ones  which,  though  differing  among 
themselves  psychologically  in  an  endless  variety  of  ways, 
are  yet  but  one  in  point  of  form,  absolutely  identical  in 
logical  frame  or  structure.  I  know  of  no  other  equal 
revelation  of  the  truly  amazing  economic  power  of  Logic 
in  our  world.  Think  of  having  to  live  in  a  world  where  no 
two  doctrines,  no  two  theories,  could  own  an  identity  of 
logical  constitution.  I  suspect  that  in  such  a  world  there 
could  be  no  logic,  no  science,  no  philosophy,  no  genuine 
life  of  intellect,  no  civilization. 

Sense  in  which  All  Doctrines  having  HaF  or  HaF'  for 
Their  Matrix  Are  Euclidean. — The  adjectives,  Euclidean 
and  non-Euclidean,  are,  as  you  are  aware,  customarily 
employed  to  designate  certain  types  of  geometry.  In 
this  use  each  of  the  adjectives  has  two  difFerent  meanings 
— one  of  them  very  specific  and  common,  the  other 
generic  and  less  common.  In  order  to  avoid  confusion 
in  reading  geometric  literature  it  is  important  to  know 
what  the  two  meanings  are.  In  its  generic  and  less  com- 
mon meaning  the  adjective  "  Euclidean "  is  used  to 
designate  the  kind  of  geometry  that  is,  in  all  important 
or  essential  respects,  identical  with  the  kind  found  in 
Euclid's  Elements.  Having  that  meaning  of  Euclidean 
in  mind,  we  should  say  that  a  given  geometry  is  non- 
Euclidean  if,  for  example,  it  is  algebraic  (or  analytic)  in 
method,  for  the  method  of  the  Elements  is  that  of  so-called 
pure  (non-algebraic)  geometry;  or  if  it  is  a  geometry  of 
four  or  more  dimensions,  for  that  of  Euclid  is  three- 
dimensional;  or  if,  like  projective  geometry,  for  example, 
or  inversion  geometry  or  the  so-called  hyperbolic  geometry 
of  Lobachevski  or  the  so-called  elliptic  geometry  of 
Riemann,  it  uses  one  or  more  postulates  inconsistent  with 
Euclid's  postulates;  or  if,  like  the  endless  series  of  geome- 


ESSENTIAL  DISCRIMINATIONS  113 

tries  (actual  or  potential)  initiated  by  Julius  Pliicker's 
great  creation  of  Line  Geometry,  it  employs  some  spatial 
entity  or  entities  other  than  the  point,  line  and  plane 
(of  Euclid's  Elements)  for  primary  element  or  elements,  or 
subject-matter.  For  all  such  distinctions  are  sufficiently 
important.  On  the  other  hand,  as  I  need  hardly  say,  a 
merely  idiomatic  or  expressional  difference, — such,  for 
example,  as  the  Greek's  saying,  "  a  straight  line  can  be 
drawn  from  every  point  to  every  point  "  whereas  we  say 
"  from  any  point  to  any  point," — is  no  warrant  for  call- 
ing the  latter  non-Euclidean.  So  much  for  the  generic 
meanings  of  the  adjectives — Euclidean  and  non-Euclidean 
as  applied  in  geometry.  And  now  let  us  be  very  clear 
as  to  what  the  specific  and  more  common  meaning  of  each 
term  is.  One  of  Euclid's  postulates — his  postulate  5 — 
had  the  fortune  to  be  an  epoch-making  statement — 
perhaps  the  most  famous  single  utterance  in  the  history 
of  science.     It  is  this: 

//   a     straight    line   falling    on    two    straight     lines 
make  the  interior  angles  on  the  same  side  less  than  two 
right  angles,  the  two  straight  lines,  ij  produced  indefinitely, 
meet  on  that  side  on  which  are  the  angles  less  than  two 
right  angles. 
Apparently  convinced  that  this  proposition  could  not 
be  deduced  as  a  theorem  from  his  other  postulates  and 
axioms,  or  common  notions,  Euclid  assumed  it.     It  was 
for  him  an  assumption,  an  hypothesis,  a  primitive  propo- 
sition, a  postulate— a  basal  proposition  of  the  Elements. 
It    is    commonly    known    as    Euclid's    parallel-postulate 
because  it  is  equivalent  to  the  postulate  that,  if  P  be  a 
point  and  L  a  line,  there  is  but  one  line  through  P  parallel 
to  L.     Unlike   Euclid,   his  successors  for  two  thousand 
years,  like  his  predecessors,  were  not  convinced  that  the 


114  MATHEMATICAL   PHILOSOPHY 

postulate  was  incapable  of  demonstration  and  many  of 
the  best  of  them  devoted  their  genius  to  vain  attempts 
at  proving  it.  At  length,  however,  mathematicians 
learned  better  and  began  to  produce  geometries  on  the 
basis  of  postulate  systems  in  which  Euclid's  parallel 
postulate  was  contradicted.  That  such  geometries  are 
just  as  logically  possible  as  Euclid's  I  will  show  in  a 
subsequent  lecture.  What  I  wish  now  to  say  is  that  any 
geometry  built  upon  a  postulate  system  containing 
Euclid's  parallel-postulate,  or  its  equivalent,  is  called 
Euclidean,  however  widely  it  may  differ  in  other  respects 
from  Euclid's  Elements ;  and,  correspondingly,  any 
geometry,  like  that  of  Lobachevski  or  that  of  Riemann, 
whose  postulate  system  contains  a  contradictory  of 
Euclid's  parallel-postulate,  is  said  to  be  non-Euclidean, 
no  matter  how  much  it  may  be  like  Euclid's  Elements 
in  other  respects.  Such  are  the  specific  and  more  usual 
senses  in  which  these  familiar  adjectives  are  employed 
in  the  literature  of  geometry. 

It  must  occur  to  you  at  once  that  there  is  no  good  rea- 
son for  confining  the  use  of  the  terms,  in  the  sense  just 
indicated,  to  geometry.  For  the  Hilbert  postulate  (13) 
being  in  agreement  with  the  parallel-postulate  of  Euclid, 
it  is  evident  that  we  may  with  evident  and  perfect  pro- 
priety call  Euclidean  all  of  the  infinitely  many  doctrines 
having  HAF  or  HAF'  for  matrix,  whether  the  doctrines 
be  true  or  false,  and  whether  they  be  geometric  or  alge- 
braic or  neither  the  one  nor  the  other. 

What  Are  the  Properties  that  a  Collection  of  Proposi- 
tional  Functions  Must  Have  in  Order  to  be  a  Postulate 
System? — I  will  name  three  properties:  pregnance,  or 
productiveness,  or  fertility;  compendence,  or  connected- 
ness;   and  compatibility,  or  consistency.     In  discussions 


ESSENTIAL  DISCRIMINATIONS  115 

of  the  question,  the  last  property  is  always  specified;  the 
first  two,  seldom  or  never,  though  they  are  evidently 
essential,  as  you  will  presently  see. 

In  saying  that  the  collection  of  propositional  functions 
must  be  pregnant,  or  fertile,  I  mean  that  it  must  be  such 
that  one  or  more  consequences,  or  theorems,  can  be 
logically  deduced  from  its  component  functions.  In 
other  words,  it  must  be  capable  of  giving  rise  to  a  doc- 
trinal function  containing  one  or  more  propositional 
functions  besides  those  serving  as  postulates.  No  one, 
I  imagine,  would  deliberately  call  a  barren  collection  of 
propositional  functions  a  postulate  system. 

In  saying  that  a  collection  of  propositional  functions, 
if  it  is  to  be  a  postulate  system,  must  be  compendent,  or 
connected,  I  mean  something  sufficiently  easy  to  grasp, 
once  it  is  perceived,  but  not  very  easy  to  state  precisely 
and  clearly.  I  will  try  to  be  intelligible.  You  know  that 
owing  to  the  presence  of  variables  in  the  postulates  of  a 
postulate  system,  the  latter  has  no  specific  subject 
matter;  we  may  say,  however,  that,  since  the  postulates 
talk  about  the  variables  as  about  subject-matter,  the 
system  has  "  apparent  "  subject-matter,  or,  better,  we 
may  say  the  system  has  undetermined  subject-matter 
represented  by  the  variable-symbols  or  variable-names. 
Now,  if  you  will  examine  the  postulates  of  some  postulate 
system,  say  those  of  the  Hilbert  system,  you  will  dis- 
cover—what you  may  not  have  before  noticed  con- 
sciously— that  the  variable-symbols  are  each  connected 
with  every  other;  that  is  to  say,  the  V\s  are  in  ^'s,  which 
are  in  &3's,  which  are  in  v^  R\  is  a  relation  of  &i's,  R2  a  rela- 
tion of  segments  composed  of  V\s  and  of  angles  composed 
of  half-rays  (or  half-&2's),  and  so  on.  This  connectedness 
gives   the   undetermined    subject-matter   of  the   system 


116  MATHEMATICAL   PHILOSOPHY 

unity,  makes  it  hang  together,  gives  it  and  the  systems 
compendence.  Suppose  you  had,  as  one  might  have,  a 
collection  of  propositional  functions  some  of  which  talked 
of  the  variables  x,  y,  z,  .  .  .  ,  and  some  of  which  talked  of 
the  variables  x'>  y',  %',  .  .  .  ,  in  such  a  way  that  the  former 
symbols  were  not  connected  with  the  latter,  then  the 
undetermined  subject-matter  of  the  collection  of  functions 
would  lack  unity;  the  collection  would  not  be  compendent 
and  would,  therefore,  not  be  a  postulate  system.  If  the 
subcollection  involving  x,  y,  z,  .  .  .  connected  them  and  if 
the  same  were  true  of  the  subcollection  involving 
x'i  y'>  z'y  '  •  •>  and  each  of  the  two  subcollections  were 
moreover,  pregnant  and  compatible,  then  the  original 
collection  would  constitute  two  postulate  systems,  but 
these  would  not  together  constitute  one.  They  would  be 
independent;  neither  of  them  being  included  in  the  other, 
they  would  not  be  related  like  the  systems  involved 
respectively  in  II AF  and  HaF',  for  the  former  of  these  is 
a  part  of  the  latter. 

In  saying  that  a  collection  of  propositional  functions, 
if  it  is  to  be  a  postulate  system,  must  be  compatible,  or 
consistent,  we  mean  that  its  functions  must  be  such  as 
not  to  involve  contradiction  among  themselves — they  will 
be  compatible  unless  at  least  two  of  them  contradict  each 
other  explicitly  or  implicitly.  The  reason  for  this  require- 
ment is  obvious.  For,  if  the  collection  contained  two 
mutually  contradictory  functions,  the  functions  of  the 
collection  would  admit  of  no  verifiers;  whatever  set  of 
admissible  constants  we  might  substitute  for  the  variables 
in  the  functions  would  yield  a  set  of  propositions  of  which 
at  least  two  would  be  mutually  contradictory;  hence,  if 
we  called  the  collection  a  postulate  system  and  then  con- 
sistently called  the  system  together  with  the  theorems 


ESSENTIAL  DISCRIMINATIONS  117 

derivable  therefrom  a  doctrinal  function,  it  would  be  one 
having  no  values,  admitting  of  no  interpretation,  giving 
rise  to  no  true  doctrine.  For  this  reason  an  incompatible 
collection  of  propositional  functions  is  not  called  a  postu- 
late system. 

If  a  collection  of  propositional  functions  be  compatible, 
how  may  the  fact  be  ascertained  ?  There  is  but  one  known 
test:  if  we  find  a  set  of  constants  that  we  are  convinced 
verify  the  functions,  then  and  only  then  we  say  the 
collection  is  compatible.  If  you  will  examine  Hilbert's 
book,  you  will  find  that  he  showed,  or  rather  indicated 
very  briefly  how  to  show,  that  his  postulate  system  is 
compatible  by  indicating  how  to  show  that  the  postulates 
are  verified  by  the  pairs  and  certain  systems  of  pairs  of  a 
specified  set  of  algebraic  numbers.  In  Lecture  VI,  I  have 
shown  in  detail  the  compatibility  of  the  postulates  of 
II AF  by  showing  that  they  are  verified  by  the  set  of  dyads 
and  certain  systems  of  dyads  of  the  real  numbers,  and 
have  indicated  the  analogous  procedure  for  the  postulates 
of  HaF'.  That  is  one  of  the  reasons  why  Lecture  VI 
is  so  detailed — I  desired  it  should  incidentally  serve  to 
exemplify  what  we  may  call  compatibility  proof. 


LECTURE  VIII 
Postulate  Properties 

SCIENTIFIC    PLATFORMS THEIR    FERTILITY,    COMPENDENCE 

AND     COMPATIBILITY DIFFERENCES     OF     EQUIVALENT 

PLATFORMS VARIETIES     OF     PLATFORMS     AND     FUNC- 
TIONS  MEANINGS   OF   INDEPENDENCE  AND   CATEGORI- 

CALNESS THEORETICAL  AND  PRACTICAL  DOUBT. 

Unless  I  am  mistaken,  you,  as  students  of  philosophy, 
should  find  no  little  interest  in  certain  questions  con- 
nected with  the  properties  I  have  mentioned  as  essential 
to  a  genuine  postulate  system  and  as  therefore  common 
to  all  such  systems.  I  desire  to  draw  your  attention  to 
some  of  the  questions  without  thereby  promising  or  pre- 
tending to  answer  all  of  them,  for  I  can  not  do  so 
satisfactorily. 

If  a  collection  of  propositional  functions  be  fertile  or 
infertile,  how  may  we  ascertain  the  fact?  Of  course,  if 
we  actually  deduce  one  or  more  consequences  from  them, 
we  then  know  that  the  collection  is  fertile.  But  the  ques- 
tion I  desire  to  ask  and  to  commit  to  you  for  future  con- 
sideration is  this:  Is  there  a  criterion  for  deciding  a  priori 
whether  a  given  collection  of  propositional  functions  is 
fertile  or  not,  and  if  there  is,  what  is  the  criterion? 

Another  question — which  I  believe  to  be  important 
and  difficult — is  this:  What  is  the  essential  nature  of  the 
role  of  variables  in  propositional  functions  by  virtue  of 

118 


POSTULATE  PROPERTIES  119 

which  a  collection  of  such  functions  has  an  ambiguous, 
or  undetermined,  subject-matter  that  may  or  may  not 
be  compendent?  May  I  leave  this  subtle  matter  to  your 
reflection? 

As  to  compatibility,  suppose  we  have  a  collection  of 
functions  such  that  we  have  not  been  able  either  to 
verify  them  or  to  prove  them  incompatible.  Doubtless, 
we  must  say,  in  such  a  case,  that  we  do  not  know  whether 
the  collection  is  compatible  or  not.  Might  not  the  col- 
lection be  incompatible  without  our  being  able  ever  to 
discover  the  fact?  Might  it  not  be  compatible,  though 
we  should  never  be  able  to  know  it  ?  Another  question — 
very  different  from  the  preceding  one — is  this:  Is  there, 
conceivably,  a  compatible  or  a  not  incompatible  collection 
of  propositional  functions  having  no  verifiers  in  our 
world  ? 

What,  essentially,  is  logical  compatibility?  Must  we 
be  content  with  mere  examples  of  it  or  with  what  seems 
at  all  events  to  be  such  examples?  Whatever  logical 
compatibility  may  be,  it  evidently  is  such  that  com- 
patibility and  incompatibility  are  related  somewhat  as 
pleasure  and  pain,  as  cosmos  and  chaos,  as  music  and 
noise,  as  health  and  disease,  as  harmony  and  discord, 
as  beauty  and  ugliness — so  that  Logic  and  Science  are  no 
less  under  the  empire  of  the  muses  than  are  the  Arts. 

Is  compatibility,  then,  an  emotion,  a  feeling,  a  mere 
sentiment?  If  it  be,  it  is  not  one  of  ideas,  but  is  a  senti- 
ment of  forms — propositional  forms.  What  is  proposi- 
tional form?  .  The  question  arose  before  and  I  said  we 
should  return  to  it.  Well,  here  it  is.  I  can  not  answer 
it.  I  know,  in  a  sense,  and  so  do  you,  what  such  form 
is,  but  I  cannot  define  it  abstractly, — not  satisfactorily. 
Possibly  you  can — sometime;    and  if  you  do,  you  will 


120  MATHEMATICAL   PHILOSOPHY 

thereby  make  a  great  contribution  to  the  science  of 
Logic.  Whatever  the  thing  may  be,  it  is  something  in 
respect  of  which,  for  example,  the  statements  "  x  is  y  " 
and  "  Socrates  is  a  man  "  are  identical;  something  in 
respect  of  which  the  statement  "  If  x  has  the  character  y 
and  whatever  has  the  character  y  has  the  character  z, 
then  x  has  the  character  z  "  is  identical  with  the  statement 
"  If  Socrates  is  human  and  whatever  is  human  is  mortal, 
then  Socrates  is  mortal  ";  something  in  respect  of  which 
either  statement  of  the  first  pair  and  either  one  of  the 
second  pair  are,  as  specimens  of  logical  material,  radically 
unlike,  irreducible  to  the  same  type. 

One  word  more  regarding  compatibility  and  I  shall 
quit  the  theme.  I  know  you  desire  to  ask, — for  in  dis- 
cussing the  matter  of  testing  for  compatibility,  students 
never  fail  to  ask,- — how  we  may  be  certain  that  we  have 
found  a  set  of  verifiers  for  a  given  collection  of  proposi- 
tional  functions.  The  answer  required  is  that  one  disci- 
plined in  the  fine  art  of  doubting  never  can  be  absolutely 
certain.  Absolute  certainty  is  a  privilege  of  uneducated 
minds — and  fanatics.  It  is,  for  scientific  folk,  an  unat- 
tainable ideal.  Perhaps  we  can  in  no  case  reach  a  higher 
degree  of  certainty  than  that  the  dyads  and  systems 
thereof,  employed  in  Lecture  VI,  satisfy  the  postulates 
there  concerned.  Yet  a  capable  doubter  may  doubt 
whether  we  sufficiently  understand  the  nature  of  the 
real  numbers  to  be  absolutely  certain  even  in  that  case. 
Even  less,  but  only  slightly  less,  difficult  is  it  to  doubt  the 
adequacy,  as  verifiers,  of  the  point  and  the  straight  line, 
though  these  have  been  used  as  such  verifiers  since  the 
memory  of  man  runneth  not  to  the  contrary.  But,  though 
we  can  never  attain  absolute  certainty  in  the  premises, 
we  can  reach  a  certainty  so  nearly  absolute  that  one  who, 


POSTULATE   PROPERTIES  121 

having  such  a  certainty  respecting  any  -practical  affair,  yet 
refrained,  for  lack  of  certainty,  from  action  where  action 
was  called  for,  would  be  rightly  judged,  not  necessarily 
stupid,  but  foolish  or  morbid  or  insane. 

The  Hilbert  Postulate  System  Not  Intrinsically  Superior 
to  Others. — In  our  discussions  this  particular  system  has 
been  the  subject  of  so  many  commentaries  that,  despite 
the  precautions  explicitly  stated  in  Lecture  II,  it  may 
seem  to  you  to  be  the  subject  of  our  study  instead  of  being 
merely  one  of  the  instruments  employed.  Perhaps  I 
need  not  remind  you  that  what  we  have  been  mainly 
examining  is  the  nature  of  the  important  general  concep- 
tion denoted  by  the  term  "  postulate  system,"  and  that, 
instead  of  beginning  with  an  abstract  definition  of  the 
concept,  we  have  preferred  to  study  it  by  means  of  a 
specific  representative,  or  typical,  example.  For  such  an 
example,  we  have  chosen  the  Hilbert  system  because  of  its 
familiarity,  accessibility  and  fame.  Our  purpose  had 
been  served  equally  well,  however,  had  we  employed 
some  other  system,  whether  logically  equivalent  or  non- 
equivalent  to  that  of  Hilbert.  Systems  of  both  kinds 
abound,  and  I  shall  presently  refer  you  to  some  of 
them. 

Equivalence  of  Postulate  Systems  and  Identity  of  Their 
Doctrinal  Functions. — Two  postulate  systems,  S"  and  S' ', 
are  said  to  be  equivalent  if,  and  only  if,  every  postulate 
in  S  is  in  S'  or  is  deducible  as  a  theorem  from  those  in 
S'  and  every  postulate  in  Sf  is  in  S  or  is  deducible  from 
those  in  S.  The  same  conception  may  be  approached 
and  viewed  as  follows:  S,  we  know,  gives  rise  to  a  doc- 
trinal function,  say,  Af,  composed  of  the  postulates  in  S 
and  the  theorems  deducible  therefrom.  Similarly,  S' 
yields  a  doctrinal   function,  say,    A/v.     Let  us  agree   to 


122  MATHEMATICAL   PHILOSOPHY 

call  AF  and  AF'  identical  if,  and  only  if,  every  proposi- 
tional  function  in  AF  or  AF'  is  in  the  other,  AF'  or  AF>  it 
being,  of  course,  understood  that  some  mere  rewording 
may  be  required  to  show  that  a  statement  in  one  is  in 
the  other.  You  see,  at  once,  that  two  postulate  systems 
are  equivalent  or  non-equivalent  according  as  the  corre- 
sponding doctrinal  functions  are  identical  or  non-identical; 
and  conversely.  It  is  a  fact  of  no  little  scientific  and 
philosophic  interest — for  it  is  far  from  "  self-evident" — 
that,  within  limits,  the  postulates  and  the  theorems  in  a 
doctrinal  function  may  interchange  their  respective  roles 
without  destroying  the  function's  identity.  Some  ques- 
tions arise  here  which,  so  far  as  I  know,  no  one  has  asked, 
and  which  I  am  unable  to  answer.  One  of  them  is: 
what  are  the  "  limits  "  within  which  the  mentioned 
interchange  of  roles  may  occur? 

The  only  way  to  know  that  two  equivalent  systems 
are  equivalent  is  to  prove  them  equivalent.  It  would  be 
very  enlightening  and  a  lot  of  fun  to  illustrate  the  process, 
but  it  would  delay  our  course  too  much.  Perhaps  you 
will  try  your  hand  at  the  game.  Two  extremely  interest- 
ing systems  which,  I  believe,  though  I  have  not  proved 
it  in  full  detail,  are  equivalent  to  Hilbert's  system  and 
consequently  to  one  another  are  the  systems  devised 
respectively  by  Professor  O.  Veblen  and  Professor  Mario 
Pieri.  Veblen's  system,  called  "  A  System  of  Axioms  for 
Geometry,"  is  found  in  Volume  V  of  The  Transactions  of 
the  American  Mathematical  Society  (1904).  This  system, 
in  a  modified  form,  was  subsequently  presented  by  its 
author  as  the  initial  monograph  in  the  Monographs  on 
Topics  of  Modern  Mathematics  (edited  by  Professor 
J.  W.  A.  Young) — a  volume  which,  though  its  articles 
differ  widely  in  aim,  spirit  and  excellence  and  though  it 


POSTULATE   PROPERTIES  123 

attempts,  pretty  successfully,  to  avoid  philosophic  ques- 
tions, may  yet  be  recommended  to  philosophical  students 
as  a  collection  of  essays  affording  an  introduction  to  a 
variety  of  important  elementary  topics  of  modern  mathe- 
matics :  namely,  Veblen's  The  Foundations  of  Geometry — 
which  does  not  deal  with  the  foundations  of  geometry  in 
general,  but  gives  the  reader  a  finely  histological  view  of 
the  ultimate  tissues  and  minute  logical  structure  of  the 
first  parts  of  Euclidean  metric  geometry;  Non-Euclidean 
Geometry,  by  Professor  F.  S.  Woods — resembling  Veblen's 
essay  in  spirit  and  method,  hardly  surpassed  as  an  intro- 
duction, for  beginners,  to  the  geometries  of  Lobachevski 
and  Riemann;  Modern  Pure  Geometry,  by  Professor  T.  F. 
Holgate — not  postulational  or  rigoristic  like  the  articles 
just  now  mentioned,  and  not  concerned  with  pure  geome- 
try in  general,  but  giving  such  an  acquaintance  with  pure 
projective  geometry  as  one  gains  of  an  immense  city  by 
riding  about  in  it  on  the  top  of  a  comfortable  'bus;  an 
exceedingly  enlightening  essay  by  Professor  E.  V.  Hunting- 
ton, dealing  postulationally  and  very  refreshingly  with 
The  Fundamental  Propositions  of  Algebra;  an  interesting 
and  instructive  article  by  Professor  G.  A.  Miller  treating 
The  Algebraic  Equation  in  part  historically,  in  part  critic- 
ally, in  a  manner  a  little  too  mature,  perhaps,  and  a  bit 
sketchy  for  beginners;  The  Function  Concept  and  the 
Fundamental  Notions  of  the  Calculus,  by  Professor  G.  A. 
Bliss — an  essay  chiefly  notable,  I  think,  as  showing  how 
swiftly  and  quickly  a  competent  reader  may  be  con- 
ducted into  the  presence  of  the  cardinal  concepts  of  the 
calculus  and  be  given  some  sense  of  their  power;  and 
three  instructive  and  stimulating  essays  by  Professors 
J.  W.  A.  Young,  L.  E.  Dickson  and  D.  E.  Smith  con- 
cerned, respectively,  with   The   Theory  of  Numbers,  Con- 


124  MATHEMATICAL    PHILOSOPHY 

structions  with  Ruler  and  Compasses  and  The  History  and 
Transcendence  of  t. 

A  moment  ago  I  referred  to  a  remarkable  postulate 
system  devised  by  the  Italian  mathematician,  Pieri.  It 
was  published  in  1899  in  Memorie  delta  R.  Academia  delle 
Scienze  di  Torino  under  the  title  "  Delia  Geometria  ele- 
mentare  come  sistema  ipotetico-detuttivo;  monografia  del 
punto  e  del  mote  ";  an  excellent  abstract  of  it  was  published 
in  1905  by  the  late  Louis  Couturat  in  his  Les  Principes 
des  Mathematiques  and  was  partly  reproduced  in  191 1  by 
Professor  J.  W.  Young  in  his  admirable  Lectures  on  Funda- 
mental Concepts  of  Algebra  and  Geometry.  I  desire,  in 
passing,  to  recommend  these  books  of  Couturat  and 
Young  as  well  worth  your  attention,  provided  you  will 
really  read  them— pondering  what  is  said  in  them — 
and  not  be  content  with  merely  glancing  through  them. 
They  handle,  in  excellent  style,  some  important  matters 
which  these  lectures  touch  but  lightly  or  not  at  all. 

You  can  not  fail  to  observe,  if  you  will  examine  and 
compare  them — as  I  hope  you  will — that  Veblen's  system 
and  that  of  Pieri  differ  from  Hilbert's  in  various  ways. 
For  example,  Veblen's  system  contains  12  postulates; 
Hilbert's,  21;  Pieri's,  20;  again,  while  Hilbert's  system 
contains,  as  we  have  seen,  five  undefined  terms,  or  vari- 
ables, Veblen's  has  but  two — "  point  "  and  "  between  " — 
and  Pieri's  also  has  but  two — "  point  "  and  "  motion." 
In  studying  the  Italian's  beautiful  system,  your  under- 
standing of  it  will  be  much  facilitated  by  noticing  that  the 
undefined  term  "  motion  "  is  used  in  the  sense  of  a 
unique  and  reciprocal  correspondence — a  one-to-one 
transformation — between  points,  and  not  in  the  man- 
in-the-street's  sense  of  a  physical  time-consuming  change 
of  place. 


POSTULATE  PROPERTIES         125 

Other  Varieties  of  Postulate  Systems  and  Doctrinal 
Functions. — By  "  other  "  varieties  I  mean  such  as  are 
not  equivalent  to  the  foregoing  systems.  Before  citing 
them,  or  rather  some  of  them — for  I  am  far  from  intending 
to  list  all  that  have  been  devised — a  word  of  caution 
seems  desirable.  In  inventing  a  postulate  system  the 
inventor  is  never,  or  almost  never,  aiming  at  the  establish- 
ment of  what  we  have  been  calling  a  doctrinal  function. 
He  is  aiming  at  establishing  autonomously  a  particular 
one  of  the  many  doctrines  which,  as  we  have  seen,  a 
doctrinal  function  has  for  its  values.  Such  special  interest 
of  the  inventor,  guiding  and  controlling  him,  is  nearly 
always  betrayed  by  the  air  or  color  of  his  speech:  often 
by  his  giving  his  system  a  kind  of  name  indicating  that 
the  system  has  a  specific  subject-matter,  which  it  has  not; 
nearly  always  by  calling  the  postulates  propositions,  which 
they  are  not,  instead  of  propositional  functions,  which 
they  are;  and  usually  by  denoting  the  undefined  terms  by 
names  instead  of  variable-symbols  as  if  the  undefined 
terms  were  constants  instead  of  variables.  This  precau- 
tion will,  I  trust,  help  to  keep  you  from  gaining  a  false 
impression  from  the  following  citations.  The  systems  in 
the  list  are  almost  random  selections,  and  the  list  is  far 
from  exhaustive  but,  by  help  of  the  numerous  references 
in  the  systems  cited,  it  will  afford  you  a  clue  to  all  or 
nearly  all  extant  systems. 

The  Axioms  of  Projective  Geometry,  by  A.  N.  White- 
head, Cambridge  University  Press,  1906. 

The  Axioms  of  Descriptive  Geometry,  by  A.  N.  White- 
head, Cambridge  University  Press,  1907. 

"  A  Set  of  Axioms  for  Line  Geometry,"  by  E.  R.  Iled- 
rick  and  Louis  Ingold,  Transactions  of  the  American 
Mathematical  Society,  Vol.  XV,  191 4. 


126  MATHEMATICAL   PHILOSOPHY 

"  On  a  Set  of  Postulates  Which  Suffice  to  Define  a 
Number-Plane,"  by  R.  L.  Moore,  Trans.  Amer.  Math. 
Soc,  Vol.  XVI,  191 5. 

"  A  Set  of  Postulates  for  Real  Algebra,  Comprising 
Postulates  for  a  One-Dimensional  Continuum  and  for 
the  Theory  of  Groups,"  by  E.  V.  Huntington,  Trans. 
Amer.  Math.  Soc,  Vol.  VI,  1905. 

"  Set  of  Independent  Postulates  for  Betweenness," 
by  E.  V.  Huntington  and  J.  R.  Kline,  Trans.  Amer. 
Math.  Soc,  Vol.  XVIII,  1917. 

"  Complete  Existential  Theory  of  the  Postulates  for 
Serial  Order,"  by  E.  V.  Huntington,  Bull.  Amer.  Math 
Soc,  Vol.  XXIII,  1917. 

"  Sui  principi  fondamentali  della  Geometria  della 
Retta,"  by  G.  Vailati,  Revista  di  Matematica,  Vol.  II,  1892. 

"  A  Set  of  Five  Independent  Postulates  for  Boolean 
Algebras,  with  Application  to  Logical  Constants,"  by 
H.  M.  Sheffer,  Trans.  Amer.  Math.  Soc,  Vol.  XIV,  191 3. 

"  Sulle  ipotesi  che  permettono  I' ' introduzione  delle  coordi- 
nati  in  una  varieta  a  piri-  dimensioni,"  by  F.  Enriques, 
Rendiconti  del  Circolo  matematico  di  Palermo,  Vol.  XII, 
1898. 

Finally,  I  will  add  to  the  foregoing  short  list  a  refer- 
ence to  the  famous  postulate  system  by  which  G.  Peano- — 
founder  and  leader  of  the  important  Italian  school  of 
workers  in  the  foundations  of  mathematics,  owning  such 
names  as  Pieri,  Padoa,  Vailati  and  others, — sought  to 
characterize  the  class  of  finite  integers.  It  is  found  in 
various  editions  of  the  Formulaire  de  Mathematiques  (as 
1899,  1901)  and  has  been  often  quoted  and  critically  dis- 
cussed— especially  by  Bertrand  Russell  in  his  Principles  of 
Mathematics,  by  Couturat  in  his  Les  Principes  des 
Mathematiques,  already  cited,  and  again,  very  recently  and 


POSTULATE   PROPERTIES  127 

illuminatingly,  by  Russell  in  his  Introduction  to  Mathe- 
matical Philosophy — a  work  which  no  student  of  philos- 
ophy can  afford  to  neglect. 

Independence  of  Postulates. — You  observe  that  in  some 
of  the  foregoing  titles  the  word  "  independent  "  occurs: 
What  does  it  signify?  It  is  used  in  a  technical  sense  and 
means  that  the  postulates  of  the  system  in  question  are 
such  that  none  of  them  is  a  logical  consequence  of  the  rest. 
As  students  primarily  of  philosophy  and  therewith  of 
epistomology,  you  should,  I  believe,  be  specially  inter- 
ested in  the  way  in  which  postulates  are  tested  for  inde- 
pendence. What  is  the  way?  It  is  this:  if  we  find  a 
set  of  constants  verifying  all  the  postulates  but  one,  then 
and  only  then  we  say  that  this  one  is  independent  of  the 
others,  for  if  it  were  not,  it  could  be  deduced  from  them 
but,  in  such  case,  it  would  be  verified  by  their  verifiers. 
If  the  system  contain  n  postulates,  then,  to  test  the 
entire  system,  it  is  obviously  necessary  to  make  n  partial 
tests — one  for  each  postulate.  If  a  collection  of  propo- 
sitional  functions  is  to  be  a  postulate  system,  is  it  essential 
that  the  functions  be  independent?  The  answer  is  no; 
independence  is  desirable  but  not  essential;  it  would  be 
essential  if  the  doctrinal  function  corresponding  to  the 
system  were  required  to  have  a  minimum  of  assumption 
and  a  maximum  of  deduction,  and  that  is  indeed  a  genuine 
ideal.  Custom,  however,  does  not  require  that  the  postu- 
lates of  a  system  be  independent;  those  of  Peano's  sys- 
tem, for  example,  are  independent;  but  those  of  Hilbert's 
are  not,  as  you  can  readily  see  by  comparing  the  last  one 
with  the  next  to  the  last.  You  will  note,  however,  that 
Hilbert  did  not  neglect  the  question  of  independence.  If 
you  disregard  postulate  (21),  his  postulates  compose  five 
sets.     He  proved  that  those  of  any  set  are  independent 


128  MATHEMATICAL   PHILOSOPHY 

of  each  other  and  that  each  set  is  independent  of  the  other 
sets. 

Categoricalness,  or  Sufficiency,  or  Completeness  of  a 
Postulate  System. — A  postulate  system  is  said  to  be 
categorical,  or  sufficient,  or  complete,  when  and  only 
when  it  has  a  certain  property  to  be  stated  presently. 
The  first  of  the  adjectives  was  introduced  into  the  litera- 
ture of  the  subject  by  Veblen,  who  received  the  suggestion 
from  Professor  John  Dewey;  the  second  one  had  been 
previously  employed  by  Huntington,  and  the  third  by 
Hilbert.  The  property  in  question,  which  a  postulate 
system  may  or  may  not  have,  is  very  interesting,  some- 
times important,  and  a  bit  subtle — not  easy  to  make 
quite  clear.  The  term  "  category,"  as  you  know,  is  a 
Greek  word  denoting  a  class,  and  we  shall  see  that  it  has 
that  meaning  here.  What  is  meant  by  categoricalness  of 
a  postulate  system?  Let  me  remind  you  that  some  of 
the  undefined  terms,  or  variables — say,  v\,  v-2,  03,  .  .  .  ,  vn 
— in  the  postulates  of  a  system  S  denote  elements,  or 
substantives,  and  the  others — say,  R\,  R2,  Rz,  .  .  .,  Rk — 
denote  relations,  or  connections  (among  the  elements). 
Let  me  further  remind  you  that,  accordingly,  a  set  of 
verifiers  of  S — a  set  of  constants,  or  meanings,  verifying 
S — is  composed  in  part  of  element-constants — the  values 
or  meanings  assigned  to  the  vs — and  in  part  of  connection- 
constants,  or  r^/^zon-constants — the  values,  or  meanings, 
assigned  to  the  R's.  Now,  in  any  given  set  of  verifiers  of 
S,  let  C\,  Co,  C3,  .  .  .  be  the  element-constants  (representing 
the  p's),  and  r\,  ro,  r%,  .  .  .  ,  the  relation-constants  (repre- 
senting the  R's);  and  in  any  other  set  of  verifiers,  let  the 
element-constants  be  c\,  cof,  C3',  .  .  .  ,  and  let  the  relation- 
constants  be  r\  ,  ro',  r%  ,  ....  If  it  be  possible  to  set  up 
a  one-to-one  reciprocal  correspondence  between  the  c's 


POSTULATE   PROPERTIES  129 

and  the  Ci"s,  the  ci  s  and  C2"s,  .  .  .  ,  in  such  a  way  that, 
if  two  or  more  c's  be  related  by  some  r,  the  corresponding 
c"s  are  related  by  the  corresponding  r',  then  and  only 
then  we  say  that  the  system  S  is  categorical,  or  sufficient. 
Two  sets  of  verifiers  that  are  transformable  into  each  other 
in  the  manner  indicated  are  said  to  be  of  the  same  type. 
It  is  easy  to  see  the  dictional  propriety  of  the  adjectives 
"  categorical  "  and  "  sufficient  "  as  thus  used.  For  if  S 
be  categorical,  or  sufficient,  it  determines  a  category,  or 
class,  of  sets  of  verifiers,  which  sets  are  all  of  them  of  the 
same  type,  and  it  (S)  is  "  sufficient  "  to  do  that.  The 
Hilbert  system  is  categorical,  as  are  some  of  the  other 
systems  above  listed.  For  an  interesting  discussion  of  the 
term  categorical,  of  the  advantages  and  disadvantages  of 
categoricalness,  together  with  detailed  proofs  that  certain 
systems  are,  and  certain  others  are  not,  categorical,  I 
may  refer  you  to  the  previously  mentioned  Fundamental 
Concepts  of  Young  and  to  Huntington's  article  in  the 
above-cited  Monographs. 

Euclid's  Postulate  System  Defective. — In  a  previous 
lecture  I  stated  that  Euclid's  postulate  system  is  defective 
— mainly  by  omission — and  promised  to  prove  the  fact 
at  a  later  stage.  Owing,  however,  to  time  limitations  and 
to  the  insistence  of  many  other  topics  which  remain  to  be 
considered,  I  have  decided  to  omit  the  proof  and  to  be 
content  with  referring  you  to  Young's  Fundamental  Con- 
cepts (pp.  12,  143)  where  the  defectiveness  in  question  is 
demonstrated  simply  and  clearly. 


LECTURE  IX 
Truth  and  the  Critic's  Art 

MATHEMATICAL   PHILOSOPHY  IN  THE   ROLE   OF   CRITIC 

A  WORLD  UNCRITICISED,  THE  GARDEN  OF  THE  DEVIL 

"SUPERSIMIAN"     WISDOM AUTONOMOUS     TRUTH 

AND    AUTONOMOUS    FALSEHOOD OTHER    VARIETIES 

OF   TRUTH   AND   UNTRUTH MATHEMATICS   AS   THE 

STUDY  OF  FATE  AND  FREEDOM ITS  PURE  BRANCHES 

AS  DOCTRINAL  FUNCTIONS ITS  APPLIED  BRANCHES 

AS  DOCTRINES THE  PROTOTYPE  OF  REASONED  DIS- 
COURSE OFTEN  DISGUISED  AS  IN  THE  DECLARATION 
OF  INDEPENDENCE,  THE  CONSTITUTION  OF  THE 
UNITED  STATES,  THE  ORIGIN  OF  SPECIES,  THE 
SERMON  ON  THE   MOUNT. 

We  have  seen  that  a  doctrinal  function  is  composed 
of  two  sets  of  propositional  functions:  an  assumed  set, 
— fertile,  compendent,  compatible,  sometimes  independ- 
ent, sometimes  categorical, — called  a  system  of  postu- 
lates; and  a  set  logically  deducible  from  the  postulates, 
and  called  theorems. 

We  have  seen  that  an  autonomous  doctrine, — a  doc- 
trine derivable  from  a  doctrinal  function  by  replacing  the 
variables  in  the  postulates  with  admissible  constants, — is 
composed  of  two  sets  of  propositions:  a  set  derived  from 
the  postulates — one  for  each  postulate;  and  a  set  similarly 
matching  the  theorems. 

330 


TRUTH  AND  THE  CRITIC'S  ART  131 

We  have  seen  that  a  doctrinal  function  is,  like  the 
propositional  functions  composing  it,  neither  true  nor 
false;  and  that  a  doctrine  derived  from  it,  is,  like  each 
of  its  component  propositions,  either  true  or  else  false. 

We  have  seen  that  a  doctrinal  function  gives  rise  to 
an  infinitude  of  true  doctrines, — values  of  the  function, — 
and  an  infinitude  of  false  ones. 

We  have  seen  that  a  doctrinal  function,  owing  to  the 
presence  of  the  variables  in  its  propositional  functions, 
has  no  specific  or  definite,  but  only  an  ambiguous  or  un- 
determined, subject-matter;  and  that,  on  the  other  hand, 
a  doctrine,  owing  to  the  presence  of  the  "substituted" 
constants  in  its  propositions,  has  a  specific,  or  definite, 
kind  of  subject-matter. 

We  have  seen  that,  in  respect  to  structure  or  form,  a 
doctrinal  function  and  all  of  the  derivable  doctrines  are 
identical,  while,  in  respect  to  content,  or  subject-matter, 
no  two  of  them  are  identical. 

We  have  seen  that,  in  the  case  of  a  doctrinal  func- 
tion, the  theorems,  (which  are  forms)  are  logically  de- 
ducible  from  the  postulates  (which  are  forms) — the 
deduction  being  purely  formal;  and  that,  in  the  case  of 
a  derived  doctrine,  the  propositions  matching  the  theo- 
rems can  not  be  logically  deduced  as  propositions  from 
the  other  propositions  as  propositions  but  only  as  forms, 
in  which  respect,  however,  the  propositions  and  the  cor- 
responding propositional  functions  are,  as  we  have  seen, 
identical;  so  that,  in  any  and  all  cases,  it  is  the  form  of 
the  premises,  and  never  their  subject-matter,  that  de- 
termines their  logical  consequences. 

Hereupon,  there  supervenes  an  important  critical 
question :  Given  a  doctrinal  function  and  one  of  the 
doctrines  derivable  from  it,  which  of  the  two  things  ought 


132  MATHEMATICAL   PHILOSOPHY 

to  be  called  a  branch  of  pure  mathematics  and  which  one 
a  branch  of  applied  mathematics?  It  is  evidently  not 
merely  a  question  of  taste,  for  the  two  things  are  not  on 
the  same  level,  they  are  not  coordinate :  the  doctrinal 
function  is  a  matrix,  the  doctrine  is  one  of  the  things  it 
moulds ;  the  former  is  form,  the  latter  has  form,  and  sub- 
ject-matter besides.  In  the  light  of  the  foregoing  con- 
spectus of  their  differences  and  similitudes,  it  is  obvious, 
I  think,  what  the  answer  must  be :  the  doctrinal  function 
is  a  branch  (or  part)  of  pure  mathematics;  the  doctrine, 
a  branch  (or  part)  of  applied  mathematics — of  what 
Lord  Bacon  called  "mixed"  mathematics,  the  mixture 
consisting  in  the  mingling  or  union  of  form  and  subject- 
matter — of  structure  and  something  having  it  or  con- 
forming to  it — of  a  prototype,  model,  mould,  or  pattern, 
and  material  owning  the  impress  thereof.  Are  we,  then, 
to  say  that  the  various  kinds  of  geometric  doctrine, — 
ordinary  Euclidean  metric  geometry,  for  example, — and 
the  various  kinds  of  algebraic  doctrine, — the  algebra  of 
the  real  numbers,  for  example, — are  all  of  them  so  many 
branches  of  applied  mathematics?  From  that  conclusion 
there  is,  I  believe,  no  escape.  They  are  quite  as  genu- 
inely, though  not  quite  so  obviously,  applied  mathematics 
as  are,  for  example,  rational  mechanics,  mathematical 
statistics,  mathematical  physics,  and  mathematical  astron- 
omy, for  the  things  which  geometries  and  algebras  are 
doctrines  about  are  just  as  genuinely,  though  less  evi- 
dently, kinds  of  subject-matter  (as  distinct  from  pure 
form)  as  are  the  things  which  the  other  mentioned 
branches  are  doctrines  about. 

In  the  view  thus  presented,  pure  mathematics  appears 
as  a  large  (potentially  infinite)  ensemble  of  doctrinal 
functions   and   applied  mathematics   as   the   ensemble   of 


TRUTH  AND  THE  CRITIC'S  ART  133 

doctrines  derivable  from  them,  having  them  for  matrices, 
and  owning  their  forms.  The  view,  as  you  will  readily 
see  upon  a  little  reflection,  is  recommended  and  confirmed 
by  its  harmony  with  many  an  insight  similarly  or  other- 
wise gained  and  usually  justified  in  other  terms. 

It  accords,  for  example,  with  the  splendid  mot  of 
Bertrand  Russell  that  "mathematics  is  the  science  in 
which  one  never  knows  what  one  is  talking  about  nor 
whether  what  one  says  is  true";  for  a  doctrinal  function, 
as  we  have  said  so  often,  has  no  determinate  subject- 
matter  and,  without  losing  its  integrity  as  a  function, 
might  conceivably  be  not  even  verifiable  by  any  of  the 
subject-matters  in  our  world. 

It  accords  with  another  just  saying  (before  quoted) 
of  the  same  author  that  "pure  logic,  and  pure  mathe- 
matics (which  is  the  same  thing),  aims  at  being  true,  in 
Lcibnizian  phraseology,  in  all  possible  worlds,  and  not 
only  in  this  higgledy-piggledy  job-lot  of  a  world  in  which 
chance  has  imprisoned  us";  for  the  connection  of  the 
theorems  of  a  doctrinal  function  with  its  postulates, — 
the  logical  lien  binding  the  former  to  the  latter  as  con- 
clusions to  premises  indissolubly,  forever, — depends  in 
no  manner  or  degree  upon  the  content,  the  accidents,  or 
the  vicissitudes  of  the  "big  buzzing  blooming  confusion" 
which  we  call  our  universe. 

It  accords  perfectly  with  the  critical  judgment,  else- 
where *  expressed,  that  "it  is  in  implications  and  not  in 
applications  that  (pure)  mathematics  has  its  lair";  for 
the  very  essence  of  a  doctrinal  function, — constituting  of 
its  elements  a  single  indestructible  Form  of  forms, — is 
that  its  postulates  logically  imply  its  theorems. 

It  accords  with  the  often  quoted  definition  of  pure 

1  Human  Worth  of  Rigorous  Thinking,  p.  303. 


134  MATHEMATICAL    PHILOSOPHY 

mathematics  given  by  Benjamin  Peirce  as  "the  science 
which  draws  necessary  conclusions" ;  for  the  theorems 
of  a  doctrinal  function  are  necessary  consequences  of  its 
postulates  in  the  sense  that  the  former  just  are  the  impli- 
cates of  the  latter. 

It  accords  with  the  judgment  of  Pieri  that  pure 
mathematics  is  a  "hypothetico-deductive"  science;  for  the 
postulates  of  a  doctrinal  function  appear  in  the  role  of 
hypotheses  and  the  theorems  in  that  of  conclusions  logi- 
cally deduced. 

It  accords  with  the  exquisite  penetrating  saying  of 
William  Benjamin  Smith  that  pure  mathematics  is  "the 
universal  art  apodictic" ;  for  the  logical  validity  of  a 
propositional  function  as  such  is  completely  independent 
of  any  and  all  particular  subject-matters,  whether  of  our 
world  or  of  any  other  that  may  be  conceivable  or  pos- 
sible, and  the  logical  coherence  of  the  theorems  and  pos- 
tulates of  such  a  function  is  apodictically  certain. 

It  accords  with  the  seemingly  shallow  but  really  pro- 
found saying  of  Henri  Poincare  that  mathematics  is  "the 
giving  of  the  same  name  to  different  things" ;  for,  despite 
the  confusion  thus  arising,  a  doctrinal  function  and  its 
various  values  are  commonly  given  a  single  name,  which 
is  usually  that  of  a  specially  important  or  familiar  one  of 
the  values. 

It  accords  well  with  the  saying  of  an  eminent  jurist 
that  "mathematics  is  the  attempt  to  seize  hold  of  God 
where  the  hair  is  shortest";  for  the  pure  forms  of  thought 
present  clean-shaven  aspects — they  are  "bald  as  the  bare 
mountain  tops  are  bald,  with  a  baldness  that  is  sublime," 
and  the  discourse  of  a  Gauss  or  a  Lagrange  is  naturally 
less  'woolly"  than  that  of  a  Cicero  or  a  Justinian  or  a 
Coke  or  a  Montesquieu  or  a  Blackstone. 


TRUTH  AND  THE  CRITIC'S  ART  135 

It  does  nqj,  however,  accord, — and  that,  too,  is  con- 
firmatory,— with  such  definitions, — no  longer  current 
among  competent  critics, — as  held  mathematics  to  be  the 
science  of  "number"  and  "space"  or  the  science  of 
"quantity"  or  the  science  of  "measurement"  or  the  "sci- 
ence of  indirect1  measurement";  for,  as  you  clearly  see, 
a  propositional  function  as  such  has  no  essential  concern 
with  such  particulars  as  number  or  space  or  quantity  or 
measurement  direct  or  indirect. 

But,  as  you  see,  it  does  accord  with  the  often  ex- 
pressed view  that  mathematics  is  the  science  of  form  and 
with  the  view  that  it  is  the  normative  science  par  excel- 
lence. 

Finally,  it  accords  perfectly  with  the  saying — reiter- 
ated many  times  and  in  may  forms  since  the  golden  days 
of  Plato — that  mathematics  contemplates  Being  under 
the  aspect  of  Eternity;  for  it  is  perfectly  clear  that  doc- 
trinal functions,  though  their  discovery  by  man  is  a 
temporal  event,  are  themselves  timeless — "older  than  the 
Sun  or  the  Sky"  and  destined  to  survive  all  things  that 
are  under  the  law  of  change  and  the  doom  of  death. 

We  have  seen  that  one  doctrinal  function  may,  with- 
out losing  its  proper  autonomy,  be  a  part  of  another  one, 
— one  autonomous  Form  of  forms  being  thus  an  integral 
constituent  of  another  such  Form  more  inclusive, — in 
which  case  any  doctrine  derivable  from  the  former  is 
similarly  a  part  of  the  corresponding  doctrine  derivable 
from  the  latter.  For  example,  H  AF  and  its  values  are 
thus  related,  as  we  saw,  to  Ha  F'  and  its  values.  Whether 
all  doctrinal  functions, — both  those  that  are  known  and 
those  that  remain  to  be  discovered, — are  somehow  logic- 

1  Auguste   Compte:    The   Positive   Philosophy    (translation   by   Harriet 
Martineau). 


136  MATHEMATICAL   PHILOSOPHY 

ally  connected  together  as  an  immense  hierarchical  gang 
of  subordinates  to  one  supreme  Function,  or  Form,  which 
embraces  the  whole  of  pure  mathematics,  is  a  question 
most  worthy  of  your  best  attention  as  students  of  phi- 
losophy. It  has  been  answered  affirmatively,  as  I  said 
in  the  introductory  lecture,  by  Whitehead  and  Russell 
in  the  Principia.  I  shall  not  here  attempt  to  justify  the 
answer  but,  for  information  regarding  the  manner  of 
the  answer  and  the  evidence  supporting  it,  I  again  refer 
you  to  that  monumental  work,  which,  as  it  is  a  composite 
of  the  most  scientific  philosophy  and  the  most  rigorous 
science,  you  will  find  a  little  harder  to  read  than  philo- 
sophical works  of  the  usual  rhetorical  type. 

The  view  I  have  been  presenting,  in  which  pure 
mathematics  appears  as  a  vast  array  of  doctrinal  func- 
tions, gives  the  science,  from  one  point  of  view,  a  pretty 
severe  aspect.  For  a  doctrinal  function  is  not  only  time- 
less, as  said,  and  indestructible,  but, — and  the  fact  merits 
our  most  pensive  meditation, — when  once  the  principles, 
or  postulates,  are  chosen,  the  die  is  cast — all  else  follows 
with  a  necessity,  a  compulsion,  an  inevitability  that  are 
absolute — we  are  at  once  subject  to  a  destiny  of  conse- 
quences which  no  man  nor  any  hero  nor  Zeus  nor  Yahweh 
nor  any  god  can  halt,  annul  or  circumvent.  Mathematics 
is,  in  a  word,  the  study  of  Fate.  Let  me  hasten  to  say 
that  the  Fate  is  not  physical,  it  is  spiritual — the  unbreak- 
able binding  thread  of  destiny  runs  through  the  universum 
of  rigorous  Thought:  the  fate  is  logical  Fate.  Is  it  a 
tyrant?  And  the  intellect,  then,  a  slave?  A  tryant  has 
whims  but  Logic  is  lawful.  Where,  then,  is  the  intellect's 
freedom?  What  do  you  love?  Poetry?  Painting? 
Architecture?  Statuary?  Music?  The  muses  are  their 
fates.     If  you  love  them,  you  are  free.     Logic  is  the 


TRUTH  AND  THE  CRITIC'S  ART  137 

muse  of  thought.  When  I  violate  it,  I  am  erratic;  if  I 
hate  it,  I  am  licentious  or  dissolute;  if  I  love  it,  I  am  free 
— the  highest  blessing  the  austerest  muse  can  give. 

The  remainder  of  this  lecture  consists  of  a  brief  dis- 
cussion of  a  very  much  neglected  subject  of  very  great 
importance;  and  its  importance  is  not  only  of  the  theo- 
retical kind  but,  as  I  trust  you  will  be  able  to  see,  of  the 
most  effectively  practical  kind  also;  practical,  that  is,  for 
such  as  have  the  talent  and  training, — the  gumption  and 
discipline, — to  employ  effectively  the  most  delicate  and 
most  powerful  of  intellectual  instruments.  I  may  call  the 
subject 

The  Role  of  Postulate  Systems  and  Doctrinal  Func- 
tions in  the  Structure  and  Criticism  of  Thought. — Here, 
as  generally  in  these  lectures,  I  use  the  term  Thought  in  a 
very  comprehensive  sense :  not  in  a  sense  so  inclusive  as  it 
has  sometimes — in  William  James's  Principles  of  Psy- 
chology, for  example,  where  it  often  signifies  or  covers 
"mental  states  at  large,  irrespective  of  their  kind";  but 
rather  in  the  sense  it  has  in  Theodore  Merz's  great  History 
of  European  Thought  in  the  Nineteenth  Century  where  the 
term  embraces  both  what  we  ordinarily  mean  by  "Science" 
and  what  by  "Philosophy";  in  other  words,  I  am  using 
the  term  Thought  to  signify  that  sort  of  discourse  which 
deliberately  owns  allegiance,  even  though  it  often  fails  in 
loyally,  to  the  authority  of  Logic.  The  subject  is,  you 
see,  immense,  penetrating  all  the  sciences  and  all  the  phi- 
losophies, natural,  or  social,  or  speculative — all  fields,  in 
short,  where  men  have  sought  by  means  of  reasoned  dis- 
course to  gain  or  to  give  wisdom  and  light  for  the  guid- 
ance of  humankind.  To  treat  it  as  it  deserves  to  be 
treated, — both  in  full  generality  and  in  detail, — would 
require  the  writing  of  a  large  volume.     Since  every  doc- 


138  MATHEMATICAL   PHILOSOPHY 

trinal  function  includes  a  postulate  system  as  its  logical 
base  and  since  pure  mathematics,  as  we  have  seen,  con- 
sists of  doctrinal  functions,  such  a  volume  might  be  ap- 
propriately entitled:  The  Role  of  Pure  Mathematics  in 
the  Criticism  of  Thought;  or,  better  perhaps,  Pure 
Mathesis  in  the  Role  of  Critic.  Possibly,  one  of  you  will 
one  day  undertake  the  production  of  such  a  work.  The 
entire  present  course  of  lectures  evidently  bears  upon  the 
task  but  the  bearing  is,  in  the  main,  implicit.  In  what 
remains  of  the  hour,  I  desire  to  discuss  the  subject,  very 
sketchily  indeed,  but  explicitly  and  in  terms.  And  I  will 
begin  with  a  word  regarding 

Autonomous  Truths  and  Autonomous  Falsehoods. — 
We  have  repeatedly  spoken  of  the  logically  organic  body 
of  propositional  functions  constituting  a  doctrinal  func- 
tion as  being  an  autonomous  form.  We  have  done  so,  as 
you  know,  because  the  thing  presents  a  certain  aspect  of 
self-sufficience  or  independence.  If  such  a  function  be 
included  in  another  one,  it  does  not  owe  its  existence,  its 
unity  or  its  integrity  to  that  relation.  It  stands  alone, 
erect,  eternal,  holding  its  principles,  its  base, — the  postu* 
lates, — within  itself,  as  it  contains  within  itself  the  logical 
lien  binding  its  elements  into  one  solitary,  self-sufficing, 
indestructible  whole.  A  doctrine  derived  from  it  (in  the 
way  now  familiar)  is  not  so  pure  as  the  function  whence 
it  was  derived;  it  is,  so  to  speak,  the  doctrinal  function 
dipt — dipt  or  immersed  in  subject-matter,  in  a  kind  of 
material  giving  each  of  the  propositional  functions  sig- 
nificance, each  of  them  thus  loaded  being  a  proposition 
and,  as  such,  true  or  false;  hence,  we  cannot  say  that  the 
doctrine  is  form  but,  as  we  have  seen,  it  has  form,  and 
the  form  it  has  is  precisely  that  which  the  doctrinal  func- 
tion is;  and  so  we  say  that  such  a  doctrine,  too,  is  autono- 


TRUTH  AND  THE  CRITIC'S  ART  139 

mous.  A  doctrine  is  the  off-spring  of  a  marriage — the 
marriage  of  subject-matter  and  pure  form;  the  latter  is 
the  mother  and  transmits  its  own  autonomy  to  all  its  chil- 
dren. If  the  doctrine  be  true,  we  may  call  it  an  autono- 
mous truth, — the  most  beautiful  and  most  precious  thing 
in  the  world, — for  it  has  the  doctrinal  function's  beauty 
of  form;  it  has  the  beauty  of  truth;  and  is,  besides,  tinged 
with  the  warmth  and  living  colors  of  some  species  of 
subject-matter  in  which  our  practical  life  is  immersed  and 
finds  its  interests  and  its  sustenance;  if,  on  the  other  hand, 
the  doctrine  be  untrue,  then  it  is  not  a  falsehood  merely, 
but  is  an  autonomous  falsehood;  this  is  indeed  not  a 
precious  thing,  it  is  the  very  opposite;  and  yet,  strange 
to  say, — for  so  pervasive  is  beauty  in  our  world, — an 
autonomous  falsehood,  despite  its  having  the  ugliness  of 
vntruth,  has  all  the  beauty  of  perfect  form — the  form 
of  the  doctrinal  function  whence  it  was  derived. 

An  autonomous  falsehood's  perfection  of  form  is 
both  a  great  advantage,  and  a  minor  disadvantage,  in 
the  quest  of  truth,  for  it  makes  it  in  one  respect  much 
easier,  and  in  one  respect  somewhat  harder,  to  detect  the 
falsity.  It  makes  it  easier,  for,  as  you  know,  an  autono- 
mous doctrine  consists  of  a  set  of  propositions  [/>],  de- 
rived from  the  doctrinal  function's  postulates,  and  a  set 
[/>'],  derived  from  the  function's  theorems;  and  hence 
two  ways, — a  direct  way  and  an  indirect  way, — are  open 
in  which  to  try  whether  the  postulates  are  satisfied:  the 
direct  way,  by  comparing  the  known  facts  in  the  field  of 
the  doctrine's  subject-matter  with  [/>]  ;  the  indirect  way, 
by  comparing  them  with  [/>'].  It  makes  it  harder,  for 
formal  perfection  is  in  itself  a  thing  so  impressive,  so 
fascinating,  so  pleasing,  that  it  tends  to  camouflage  a  de- 
fect of  content  and  thus  to  deceive  by  a  kind  of  agreeable 


140  MATHEMATICAL   PHILOSOPHY 

dazzling  of  the  mind.  The  disadvantage  in  question, 
though  always  inferior  to  the  mentioned  advantage,  is 
naturally  more  serious  in  cases  where  the  question  of 
postulate  verification  is  especially  difficult  to  answer  with 
perfect  certitude.  Such  cases  are  not  merely  supposable; 
they  are  in  fact  of  very  frequent  occurrence  in  the  history 
of  science.  Just  at  present,  we  have  indeed  a  living  il- 
lustration in  the  world-wide  discussion  of  relativity 
theories,  wherein  the  satisfiedness,  or  verifiedness,  of 
certain  famous  postulates  (or  deductions  therefrom)  — 
once  regarded  as  established,  long  so  regarded  in  the 
case  of  some  of  them — has  been  called  in  question  and  is 
now  held  in  doubt  or  denied.  For  a  presentation  of  the 
great  matter  of  these  theories,  I  have  real  pleasure  in 
referring  you  to  C.  D.  Broad's  article,  "Euclid,  Newton, 
and  Einstein,"  in  The  Hilbert  Journal,  Vol.  XVIII. , 
April,  IQ20;  the  article,  which  is  easily  the  best  I  have 
seen  on  the  subject,  is  quite  notable  as  a  sound,  intelligible, 
semi-popular  exposition  of  an  exceedingly  recondite  sci- 
entific development.1  The  art  of  such  exposition,  let  me 
say  in  passing,  is  difficult  and  important — quite  as  dif- 
ficult and,  in  its  service,  quite  as  important  as  research 
itself;  a  high  degree  of  skill  in  it  is,  I  think,  not  less 
rare  than  a  high  degree  of  research  ability;  once  in  a 
great  while,  the  two  things  are  united  in  one  personality, 
as  in  W.  K.  Clifford,  for  example,  in  Thomas  Huxley, 
in  Helmholz  and  Ernst  Mach,  but  not  in  Henri  Poincare 
who,  though  he  repeatedly  essayed  the  task  of  popular 
exposition  and  indeed  produced  many  a  lightning  flash 

1  Since  this  was  written  many  attempts  have  been  made  to  explain  the 
doctrine  in  popular  terms.  Among  the  best  attempts  may  be  mentioned 
Bolton's  Introduction  to  the  Theory  of  Relativity  [E.  P.  Dutton  &  Co.] 
and  W.  B.  Smith's  article,  Relativity  and  Its  Philosophic  Implications 
[Monist,  Dec,  1921]. 


TRUTH  AND  THE   CRITIC'S  ART  141 

in  the  layman's  sky,  yet  lacked  the  requisite  patience  for 
continuous  clarity. 

Heteronomous,  or  Anantonomous,  Doctrines,  True 
and  False. — Everyone  knows  that  each  of  the  great  sub- 
jects belonging  to  the  domain  of  Thought  has  a  more  or 
less  reasoned  literature, — often  an  immense  literature, — 
of  its  own.  Everyone  knows  that  any  such  literature, — 
the  literature  of  any  such  subject, — is  composed  of  a 
number  of  more  or  less  logically  organized  bodies  of 
propositions.  It  is  common  and  convenient,  as  everyone 
knows,  to  speak  of  such  a  body  of  propositions, — no 
matter  what  the  subject, — as  a  theory  or  a  philosophy  or 
a  science  or  a  doctrine.  Let  us  here  employ  the  term 
last  mentioned.  Everyone  knows  that  in  every  great 
subject  such  doctrines  are  not  only  numerous  but  that, 
by  modification  of  old  ones  and  addition  of  new  ones, 
the  number  is  constantly  increasing.  Together  they  con- 
stitute our  more  or  less  reasoned  wisdom, — what  Clar- 
ence Day  would  call  our  supersimian  wisdom, — about  the 
world. 

I  desire  to  draw  your  attention  to  the  fairly  obvious 
fact  that  most  doctrines, — the  vast  majority  of  doctrines 
whether  true  or  false, — are  not  autonomous.  Autonomy, 
— the  quality  of  being  autonomous, — is  an  ideal;  it  is  an 
ideal  to  which  doctrines  in  every  subject,  or  the  builders 
of  them,  do  indeed  more  or  less  consciously  aspire  and 
to  which  they  slowly,  for  the  most  part  very  slowly,  ap- 
proximate but  which  they  seldom  even  nearly  attain. 
When  a  doctrine  does  reach  (or  nearly  reach,  for  it  can 
not  quite  reach)  the  ideal,  when  it  attains  close  approxi- 
mation to  autonomy,  then  and,  strictly  speaking,  only 
then  it  has  become  mathematical;  the  immense  majority 
of    doctrines    are,    then,    non-mathematical,    lacking    au- 


142  MATHEMATICAL  PHILOSOPHY 

tonomy.  They  are  heteronomous,  or  anautonomous,  doc- 
trines. These  are,  I  grant  you,  terrifying  words.  Why 
not  say  mathematical  and  non-mathematical,  and  have 
done  with  it?  Because  the  other  words  serve  to  direct 
and  fix  attention  upon  what  is  precisely  characteristic  of 
mathematical  doctrine,  on  the  one  hand,  and  of  the  non- 
mathematical,  on  the  other. 

Why  is  it  that  nearly  all  doctrines  in  the  world,  even 
those  which  deal  with  the  most  familiar  subjects,  not 
excluding  some  doctrines  that  are  currently  called  mathe- 
matical, have  been  and  are  tfwautonomous?  The  causes 
are  evidently  many.  There  is  the  general  feebleness,  the 
logical  meagreness,  of  the  human  intellect;  there  are  the 
strong  unruly  passions  of  men  driving  them  in  uncharted 
courses  as  rudderless  vessels  in  a  storm;  there  are  their 
lusts  and  greeds  aiming  at  the  gratification  of  propensi- 
ties infinitely  beneath  and  commonly  hostile  to  the  craving 
for  truth;  there  are  laziness,  fickleness,  and  impatience; 
there  is  the  marvelous  copiousness  and  prodigality  of 
mother  Nature  enabling  her  children  to  get  on  somehow 
even  though  they  have  but  meagre  care  for  wisdom;  and, 
finally,  there  is  the  inherent  intractableness  of  the  great 
subject-matters  with  which  most  doctrines  deal. 

Hence  a  rough  general  answer  to  our  question  evi- 
dently is  that  the  building  of  an  autonomous  doctrine  re- 
garding any  great  matter  is,  for  us  humans,  constituted 
and  circumstanced  as  we  are,  exceedingly  difficult,  while 
the  making  of  the  other  kind  is  easy :  there  are  so  many, 
many  ways  in  which  a  doctrine  may  fail  of  autonomy — 
so  many  possibilities,  so  many  opportunities,  so  many  so- 
licitations from  within  ourselves  and  from  without,  for 
going  wrong  in  the  business  and  incurring  delay.  Do 
but  reflect  a  little  upon  the  matter.    An  autonomous  doc- 


TRUTH  AND  THE  CRITIC'S  ART  143 

trine,  we  have  seen,  is  one  derivable  from  a  doctrinal 
function  and  inheriting  its  form.  Today  indeed  we  are 
familiar  with  the  general  conception  of  such  functions  and 
have  numerous  examples  of  it;  and  we  are  in  some  danger 
of  inferring  or  supposing  that  the  formation  of  that  con- 
ception and  the  discovery  of  the  known  functions  exempli- 
fying it  have  been  accomplished  easily.  But  such  an  in- 
ference or  supposition  would  be  very  erroneous.  Those 
whom  we  conventionally  call  the  authors  of  the  known 
doctrinal  functions  are  not,  strictly  speaking,  their  dis- 
coverers. Far  from  it.  The  discovery  of  them  and  of 
the  general  concept  they  exemplify  is  not  the  achievement 
of  an  individual  but  of  a  very,  very  long  series  of  indi- 
viduals; it  is,  like  all  other  forms  of  wealth,  like  all  other 
elements  of  civilization,  a  racial  achievement — the  slowly 
accumulated  fruit  of  many  generations  of  dead  men's  toil. 
And  clear  consciousness  of  the  outcome, — of  the  fact  and 
nature  of  the  fruit, — is  of  very  recent  date.  To  realize 
vividly  that  such  is  the  case,  you  need  only  reflect  that 
doctrinal  functions  are  composed  of  propositional  func- 
tions and  that,  as  we  saw  in  a  previous  lecture,  the  su- 
premely important  notion  of  propositional  function  came 
to  recognition  and  received  a  name  only  a  few  years  ago. 
Compared  with  the  vast  backward  stretch  of  human  time, 
— say,  a  quarter  or  a  half  million  years, — the  interval 
from  Euclid's  day  to  ours  is  indeed  very  short;  virtually 
we  are  among  Euclid's  contemporaries;  yesterday  he  was 
here;  yet  his  Elements  is  our  human  race's  earliest  ex- 
ample of  a  doctrinal  function  and  even  it  is  an  imperfect 
example,  failing,  as  we  have  seen,  to  state  certain  of  the 
postulates  explicitly,  and  being  in  appearance,  as  he  prob- 
ably conceived  it  to  be  in  fact,  a  specific  doctrine  instead 
of  a  doctrinal  function. 


144  MATHEMATICAL    PHILOSOPHY 

But  the  sheer  difficulty  of  attaining  doctrinal  au- 
tonomy, great  as  the  difficulty  is,  is  by  no  means  solely 
responsible  for  the  fact  that  so  few  of  the  doctrines  in 
the  world  are  autonomous  and  that  nearly  all  the  rest 
of  them  are  very  remote  from  that  estate.  Part,  a  very 
large  part,  of  the  explanation  is  found  in  the  fact, — 
abundantly  manifest  in  the  history  of  thought  and  for 
most  of  us  strongly  confirmed,  I  fear,  by  our  introspective 
knowledge  of  ourselves, — that  we  humans  in  our  doctrinal 
constructions  and  preachments  are  but  seldom  much  con- 
cerned to  make  them  even  approximately  perfect  in  re- 
spect to  logical  form;  in  their  content  our  interest  is,  in 
general,  far  greater;  but  even  as  to  content,  we  have,  I 
think,  to  own  that  we  are,  in  general,  much  less  concerned 
to  have  our  doctrines  ultimately  true  than  to  have  them 
instantly  elective.  I  trust  I  am  not  sufficiently  depraved 
to  believe  in  the  total  depravity  of  man;  for  many  of  his 
supersimian  traits  and  for  some  of  his  simian  qualities, 
I  have  profound  admiration;  but  in  candor  we  must  own, 
I  believe,  that  wholly  disinterested  pursuit  of  truth  is  very 
rare.  We  humans  desire  indeed  to  be  regarded  devoted 
lovers  of  truth  and  we  flatter  ourselves  that  we  are  such 
in  fact;  sometimes  we  are,  but,  in  general,  we  are  not; 
in  general,  we  prefer  something  else;  we  often  boast  that 
we  are  not  theoreticians,  and  the  boast  has  its  basis  in  fact; 
we  are  not  theoreticians,  we  are  practicians,  though  we 
dislike  the  word  and  call  ourselves  practical  instead. 
Being  primarily  and  predominately  practical  in  our  in- 
terests, when  we  are  building  doctrines,  though  we  al- 
ways pretend  to  be  thus  endeavoring  to  set  forth  truth, 
we  are,  with  rare  exceptions,  animated  by  a  very  different 
motive;  we  are  not  trying  to  formulate  something  that, 
by  painstaking  research,  we  have  found  to  be  true;  we 


TRUTH  AND  THE  CRITIC'S  ART  145 

are,  instead,  though  we  do  not  confess  and  may  not  even 
know  our  real  motive,  trying  to  make  an  instrument  that 
will  "work,"  that  will  be  an  effective  means  to  some  prac- 
tical end;  we  are  not, — however  much  we  pretend  to 
be, — endeavoring  to  enlighten  our  fellow  men — we  are 
endeavoring  to  influence  them :  our  aim  is  not  the  ad- 
vancement of  wisdom;  it  is,  in  current  slang,  to  put  some- 
thing over  or  across.  And,  as  already  intimated,  the 
dominance  of  this  motive  is  entirely  consistent  with  sin- 
cerity. The  builder  or  the  advocate  of  a  doctrine  os- 
tensibly aiming  at  truth  but  really  aiming  at  some  prac- 
tical achievement,  may  be  entirely  sincere — he  may  indeed 
be,  like  Mahomet,  for  example,  like  Deacon  Paris  or 
Lenin,  a  fanatic,  incapable  of  doubt,  incapable  (that  is) 
of  doubting  the  validity  or  justice  of  his  central  thesis, 
and  hence  incapable  of  scientific  devotion  to  truth. 

Now,  it  is  evident  that  one  making  or  advocating  a 
doctrine,  if  he  be  animated,  not  by  the  genuine  philoso- 
pher's love  of  truth,  but  by  the  spirit  of  the  partisan  and 
propagandist,  if  he  have  not  the  disinterestedness  of  the 
genuinely  scientific  worker  but  have  instead  the  interest 
of  one  bent  on  driving  through  to  the  goal  of  some  prac- 
tical purpose  by  any  and  every  available  means  thereto 
— it  is  evident,  I  say,  that  such  a  one  will  not  desire  to 
bring  his  doctrine  to  the  perfection  of  logical  form  but 
will  often  indeed  desire  the  very  opposite;  and  the  rea- 
sons are  plain:  to  make  a  doctrine  autonomous  requires 
much  patience  and  time,  but  the  practician,  the  partisan, 
the  propagandist,  is  by  nature  impatient — he  is  eager  for 
results;  in  trying  to  make  a  doctrine  autonomous,  we 
usually  discover  that  the  doctrine  is  false  (for  most  doc- 
trines are  false),  but  such  a  discovery,  which  tends  to 
dampen  ardor,  is  just  what  your  partisan  or  your  fanatic 


146  MATHEMATICAL   PHILOSOPHY 

most  desires  to  avoid;  an  autonomous  doctrine,  because 
its  elements  are  arranged  in  order  and  their  logical  con- 
nections are  bared,  is  thereby  prepared  for  relatively  easy 
examination  by  others,  so  that,  if  the  doctrine  be  false, 
the  fact  is  specially  liable  to  detection,  but  it  is  not  the 
aim  of  your  propagandist  to  make  such  detection  easy; 
a  doctrine,  once  it  is  made  autonomous,  though  it  has 
thus  gained  in  light,  has  lost  its  heat,  it  is  lacking  in  punch, 
as  we  say,  or  "pep,"  it  is  prepared  for  the  service  of  mere 
enlightenment;  but  your  propagandist,  your  fanatic  and 
your  partisan  do  not  seek  to  enlighten,  they  seek  to  in- 
fluence,— to  get  action, — and  so  they  keep  their  doctrines 
amorphous,  malleable,  and  charged  with  emotion  for 
emotional  utterance  and  emotional  effect. 

Well,  you  may  say,  what  is  to  be  done  about  it? 
What  is  the  remedy?  The  remedy  is — Criticism — the 
Gadfly:  patient,  unsparing  logical  criticism  of  one's  own 
work  in  doctrine  building;  and,  in  all  subjects,  keen, 
merciless,  stinging,  gadfly  criticism  of  any  and  all  half- 
baked,  logically  amorphous,  flabby  doctrines  pretending 
to  be  important  embodiments  of  truth  or  vessels  of  wis- 
dom. Men  must  be  driven  by  art, — the  art  of  criticism, 
— to  levels  of  excellence  higher  than  those  to  which  they 
are  drawn  by  unenlightened  nature. 

I  am,  I  hope,  not  misunderstood  in  this  matter.  I 
am  far  from  contending, — no  one  can  be  so  foolish  as  to 
contend, — that  in  every  field  of  thought  workers  can  be 
constrained  by  criticism  to  put  their  results  in  the  logically 
perfect  form  of  an  autonomous  doctrine;  man  can  not 
be  constrained  to  perform  the  impossible  nor  to  do  in- 
stantly what  has  at  best  a  very  remote  possibility  of  being 
done  at  all :  what  I  do  contend  is  that  in  all  departments 
of  thought  men  can  be  constrained  by  criticism  to  have 


TRUTH  AND  THE  CRITIC'S  ART  147 

constant  regard  to  the  principles  and  the  spirit  of  mathe- 
matics— the  spirit  of  dispassionate  thought — to  estimate 
the  logical  cogency  of  their  thinking  in  accordance  with 
mathematical  standards,  to  employ  the  postulational 
method  in  many  instances  where  it  has  never  been  em- 
ployed nor  even  attempted,  to  hold  it  as  a  model  in  all 
cases,  and  in  all  their  work  to  own  the  authority,  even 
though  they  can  not  attain  the  perfection,  of  the  doctrinal 
function  as  the  highest  and  purest  of  logical  ideals;  and 
I  contend  that,  if  this  were  done,  both  the  logical  quality 
and  the  /rz^/z-quality  of  what  I  have  called  "the  more 
or  less  reasoned  literature"  of  the  world  would  be  thereby 
rapidly,  constantly  and  immeasurably  improved. 

Finally,  I  would  direct  your  best  attention  to  the  fact 
that  everywhere  in  that  literature, — the  literature  of 
Thought, — there  are  to  be  found  certain  phenomena, 
certain  common  characters,  which  invite  us  to  the  indi- 
cated type  of  criticism  as  to  a  great  and  hopeful  enter- 
prise. What  I  mean  is  this:  if  you  will  select  any  well- 
known  doctrine,  no  matter  how  amorphous,  belonging  to 
any  field,  no  matter  how  remote  or  seemingly  remote 
from  mathematics — it  may  be  in  natural  science  or  in 
philosophy  or  in  theology  or  in  ethics  or  in  law  or  in 
education  or  in  politics  or  in  economics  or  in  history  or 
in  sociology  or  in  education — if  I  say,  you  will  select  from 
any  such  field  a  doctrine  worthy  of  attention  and  examine 
it,  you  will  find  that  the  author  has  more  or  less  con- 
sciously recognized,  in  at  least  some  small  measure,  the 
necessity  of  working  with  principles  which  he  may  not 
have  explicitly  stated  as  such  either  in  whole  or  even  in 
part;  you  will  find  that  he  has  consciously  or  unconsciously 
made  use  of  certain  (or  uncertain)  primitive  propositions 
or  propositional  functions, — certain  assumptions,  that  is, 


148  MATHEMATICAL    PHILOSOPHY 

or  postulates, — which  he  may  or  may  not  have  regarded, 
and  may  or  may  not  have  recorded,  as  such;  you  will 
probably  find  that  he  has  tried  to  define  certain  of  his 
terms  and  will  certainly  find  that  other  terms  ostensibly 
defined  or  not,  have  in  fact  been  virtually  employed,  de- 
liberately so  or  not,  as  undefined  terms  (primitives,  or 
variables)  ;  you  will  find  that  he  has  stated  a  series  of 
propositions  which  he  has  made  some  effort  to  prove, 
to  demonstrate,  to  deduce,  by  a  process  of  reasoning, 
from  something  or  other;  in  a  word,  you  will  find  that 
within  the  doctrine,  however  formless,  however  ill  or- 
dered its  parts,  however  loosely  knit  its  texture,  there  is 
shadowed  forth  more  or  less  clearly,  very  dimly  it  may 
be,  something  of  the  figure  and  frame  of  the  logical  pro- 
totype called  doctrinal  function,  as  if  this  thing  were  so 
built  into  the  very  constitution  of  intellect  as  in  some 
measure  to  guide  and  shape  its  activity  whether  we  will 
or  no. 

It  would  amply  compensate  us  for  the  toil  involved, 
had  we  the  time  for  it,  to  devote  one  or  more  lectures 
of  this  course  to  illustrating  the  truth  of  what  I  have  just 
said  by  critically  examining  one  or  more  outstanding  doc- 
trines of  the  non-mathematical,  or  anautonomous,  sort 
with  a  view  to  discovering  the  presence  in  them  of  the 
mentioned  phenomena.  But,  except  for  a  few  hints  to  be 
presently  given,  I  must  leave  the  task  for  you,  commend- 
ing it  as  being  in  my  judgment  the  best  possible  discipline 
in  the  great  art  of  doctrinal  criticism,  for  which  the  pres- 
ent condition  of  the  wrorld  calls  more  loudly  than  ever 
before  and  which  it  is  your  supreme  privilege  and  supreme 
duty  as  philosophers  to  master,  foster,  and  practise. 

You  have  the  clue  and  the  material  abounds  on  every 
hand.     "I  do  not  frame  hypotheses"    (Hypotheses  non 


TRUTH  AND  THE  CRITIC'S  ART  149 

jingo)  said  Newton  and  he  accordingly  called  his  prin- 
ciples of  dynamics  "Axioms  or  Laws  of  Motion"  (Axio- 
mata  sive  leges  motus).  Today,  however,  even  we,  who 
are  hardly  Newtons,  know  that  his  "axioms"  are  not 
absolute  certitudes,  that  they  are  not  self-evident  propo- 
sitions, that  they  are  indeed  not  perfectly  clear;  we  know 
that  they  are  "hypotheses,"  pure  assumptions,  postulates, 
genuine  propositional  functions  in  which  the  variables 
are,  in  Euclidean  fashion,  denoted  by  names,  which  names 
or  some  of  them  are,  again  in  Euclidean  fashion,  "de- 
fined"— defined  by  definitions,  or  descriptions,  serving 
merely  to  indicate  one  specific  interpretation  of  the  func- 
tions. For  a  good  approximation  to  the  sort  of  criticism 
I  am  recommending,  let  me  refer  you  to  an  examination 
of  Newton's  doctrine  of  dynamics  by  the  late  Ernst  Mach 
in  his  masterful  Science  of  Mechanics.  Do  you  wish  to 
say  that  this  doctrine  is  mathematical?  Very  well,  it  is 
mathematical  but  it  is  not  purely  such  and  I  have  cited 
it  partly  on  that  account  and  partly  because  of  its  fame. 
Let  us,  however,  take  a  glance  in  other  directions.  Con- 
sider, for  example,  that  most  significant  of  all  American 
political  documents — The  Declaration  of  Independence. 
It  is,  or  contains,  in  epitome,  a  political  doctrine  of  the 
highest  importance.  In  saying,  "We  hold  these  truths  to 
be  self-evident,"  its  authors  virtually  said,  JFe  lay  down 
the  following  postulates;  and  the  list  they  give  of  "self- 
evident  truths"  is  clearly  a  list  of  their  political  postu- 
lates. These  are  propositional  functions;  a  little  scrutiny 
will  enable  you  to  detect  the  undefined,  or  variable,  terms, 
which  the  authors  of  course  assumed  would  be  understood 
in  some  specific  sense.  The  postulates,  you  observe,  are 
swiftly  followed  by  important  deductions.  I  can  not  here 
further  elaborate  the  matter,  but  you  can  not  fail  to  de- 


150  MATHEMATICAL   PHILOSOPHY 

tect  in  it  the  outlines  or  rudimentary  presence  of  a 
singularly  impressive  doctrinal  function  of  political  type 
and  to  feel  invited  to  examine  the  great  document  and 
perhaps  to  elaborate  it  in  accordance  with  the  conception 
and  method  of  such  functions. 

For  another  example,  consider  the  Constitution  of  the 
United  States.  It  may  be  regarded  in  the  same  light, 
only  to  do  so  requires  a  little  more  ingenuity.  Omit,  "We 
the  people  of  the  United  States  do  ordain  and  establish 
this  Constitution  for  the  United  States  of  America"; 
what  is  left  embodies  a  doctrine — a  doctrine  in  the  field 
of  government.  What  are  its  postulates?  Everything 
from  article  I  to  the  end  of  the  document — of  course,  the 
provisions  are  not  stated  in  the  manner  of  postulates  but 
they  can  be  so  stated.  Where  are  the  theorems?  These 
are  not  stated  at  all  but  are  involved  in  the  meanings  of 
the  great  phrases  respecting  justice,  tranquillity,  and  so 
on  of  the  heavily  laden  preamble.  I  wonder  if  what  I 
have  said  is  a  sufficient  hint.  The  doctrine  in  question  is, 
in  a  word,  this:  the  provisions  in  the  Constitution, — that 
is,  the  postulates, — imply  the  body  of  unstated  proposi- 
tions involved  in  the  great  terms  of  the  preamble.  Such, 
in  a  nut-shell,  was  the  thought,  the  doctrine,  of  the  fathers. 
Let  me  offer  a  similar  hint,  a  mere  hint,  regarding  the 
philosophy  of  Descartes.  You  know  how  strenuously  he 
sought  for  a  basis  of  indubitable  fact.  He  was  seeking, 
though  he  did  not  so  conceive  the  task,  for  the  indubit- 
able verification  of  certain  propositional  functions,  which 
he  did  not  indeed  formulate  nor  evoke  from  the  shadowy 
background  of  his  thought.  One  of  the  verifications  he 
found  or  thought  he  found  is,  as  you  know,  of  world 
wide  and  immortal  fame.  Consider  the  propositional 
function:  if  x  performs  a  kind  y  of  activity,  then  x  has 


TRUTH  AND  THE  CRITIC'S  ART  151 

the  property  z.  Let  x  denote  /;  let  y  denote  thinking,  let 
z  denote  being,  drop  the  hypothetical  form,  and  you  have 
Descartes's  Cogito,  ergo  sum;  Je  pense,  done  je  suis;  1 
think,  therefore  I  am.  Enough  of  hints.  The  suggested 
type  of  analysis  is  evidently  applicable  on  every  hand — 
to  the  Sermon  on  the  Mount,  to  the  Republic  of  Plato,  to 
Darwin's  Origin  of  Species,  to  the  League  of  Nations 
Covenant,  to  Marxian  Socialism,  to  the  Soviet  Constitu- 
tion of  Russia,  to  the  Constitution  of  the  German  Re- 
public, to  the  Einstein  Doctrine  of  Relativity,  to  the  Bry- 
anistic  Ethics  of  Prohibition — to  all  manner  of  doctrin- 
istic  contentions  of  wise  men,  knaves,  fanatics  and  fools. 

The  type  of  criticism  I  am  here  advocating  and  urg- 
ing as  supremely  important  shapes  itself,  as  you  see,  very 
simply.  Confronted  by  a  doctrine  in  any  department 
of  thought,  Criticism  demands  answers  to  these  questions : 
What  is  assumed — what  are  the  postulates?  What  are 
the  undefined,  or  variable,  terms?  What  are  the  theorems 
or  proved  propositions  and  what  the  defined,  or  constant, 
terms?  How  have  the  theorems  been  deduced,  and  the 
defined  terms  defined?  What  meanings  have  been  as- 
signed to  the  variable  terms,  and  how?  Upon  these  ques- 
tions, criticism,  if  it  is  to  be  criticism  of  Thought,  is  bound 
to  insist — there  is  no  alternative.  Such  criticism  is  a  civi- 
lizing agency — the  guardian  of  the  principles  of  freedom. 
Without  it,  the  world  becomes  a  wilderness  of  error  and 
lust — the  garden  of  the  Devil. 

Easy  to  ask,  the  questions  are,  in  general,  not  easy 
to  answer,  and  the  difficulty  of  answering  rightly  is  usually 
greatest  just  where  it  is  most  important  to  compel  an 
answer — in  the  case,  that  is,  of  amorphous,  emotion- 
charged  "dynamic"  doctrines  that  pretend  to  aim  at  en- 
lightenment but  really  aim  at  victory  and  win  it  by  ap- 


152  MATHEMATICAL   PHILOSOPHY 

pealing,  not  to  love  of  truth,  but  to  lust  for  power  or 
gain.  If  the  author  be  unable  to  answer,  criticism  must 
drive  him  back  to  the  silence  of  the  cloister  for  further 
study.  If  he  contend,  as  sometimes  he  will  contend,  that 
he  has  defined  all  his  terms  and  proved  all  his  proposi- 
tions, then  either  he  is  a  performer  of  logical  miracles 
or  he  is  an  ass;  and,  as  you  know,  logical  miracles  are  im- 
possible. 

Allow  me,  in  closing,  an  additional  word  to  guard 
against  a  possible  misapprehension.  We  have  seen  and 
said  that  a  doctrine,  in  becoming  autonomous,  though  it 
thus  gains  in  light,  loses  in  heat — it  tends  to  become 
static.  Is  it  not  true,  however,  that  to  do  its  work  in  the 
world  a  doctrine  must  be  dynamic?  The  answer  is,  it 
must;  and  nothing  has  been  said  to  the  contrary.  It  is 
necessary  to  distinguish:  to  test  for  truth  is  one  thing;  to 
utter  is  another.  Of  these  two  things,  the  former  is  the 
duty  to  which  men  must  be  driven  by  criticism  if  they 
be  not  drawn  to  it  by  love  of  truth.  Once  the  test  is 
made  and  the  doctrine  found  not  wanting,  then,  and  not 
before,  it  may  be  legitimately  urged  home  with  full  ardor 
by  all  the  arts  of  utterance  even  though  the  truth  be  thus 
made  to  burst  upon  us  like  the  thunder  of  Wagnerian 
music,  making  the  mountains  tremble,  the  seas  vibrate, 
and  seeming  to  shake  the  very  rafters  of  the  sky. 


LECTURE  X 
Transformation 

NATURE      OF      MATHEMATICAL      TRANSFORMATION NO 

TRANSFORMATION,  NO  THINKING TRANSFORMA- 
TION LAW  ESSENTIALLY  PSYCHOLOGICAL RELA- 
TION    AND     FUNCTION     AND     TRANSFORMATION     AS 

THREE    ASPECTS    OF    ONE    THING ITS    STUDY    THE 

COMMON     ENTERPRISE     OF    SCIENCE THE     ART     OF 

MATHEMATICAL    RHETORIC THE    STATIC    AND   THE 

DYNAMIC     WORLDS THE     PROBLEM    OF    TIME     AND 

KINDRED    PROBLEMS IMPORTATION    OF    TIME    AND 

SUPPRESSION  OF  TIME  AS  THE  CLASSIC  DEVICES  OF 
SCIENCES 

Looking  back  to  the  days  of  my  youth,  I  see  pretty 
clearly  and  a  little  sadly  that  in  this  good  land  of  ours 
secondary  and  collegiate  mathematical  instruction  was, 
with  little  exception,  then  remarkable  for  two  things :  its 
emphasis  and  its  silence.  It  was  very  diligent  and  very 
emphatic  about  small  matters;  about  great  ones  it  was 
dumb.1  I  am  led  to  this  reflection  by  recalling  my  first 
and  second  introductions  to  the  term,  "transformation." 
The  first  was  in  algebra;  there  was  a  chapter  on  the  cubic 
and  the  biquadratic  equations,  which  we  were  to  learn 

'Sec    in    this    connection    Professor    J.    C.    Fields'    brilliant    address- 
Universities,  Research  and  Brain  Waste — published  by  the   University  of 
Toronto  Press,   1920. 

153 


154  MATHEMATICAL   PHILOSOPHY 

how  to  solve ;  in  the  chapter  was  a  section  headed,  Trans- 
formation of  Equations;  in  it  the  x  of  our  equations  was 
replaced  by  something  else;  so  we  got  new  equations; 
these  were  managed  so  that,  after  a  few  "stunts,"  we 
had  the  roots  sought;  just  what  part  of  the  proceeding 
was  dubbed  transformation  and  what  not,  we  were  left  to 
conjecture,  but,  in  those  days,  even  conjecture, — guessing, 
— was  a  kind  of  sin,  for  mathematics  was  "the  exact  sci- 
ence," it  was  just  "pure  reasoning."     I  wonder  if  you, 
who   are   of   a   later   generation,   were   more   fortunate. 
Well,  my  second  introduction  to  the  term  in  question  oc- 
curred in  analytical  geometry,   for  the  book  contained, 
over  toward  the  middle  of  it,  an  arid  little  chapter  en- 
titled  Transformation  of  Coordinates — a  meagre,   dull, 
stupid,    stupefying    parched    little    desert    discussion,    of 
which  no  use  was  made  there  and  very  little  in  the  subse- 
quent part  of  the  course.     In  neither  of  the  "introduc- 
tions" was  there  an  illuminating  word  by  text-book  or 
by  instructor  to  signalise  the  significance  of  the  matter  in 
hand — no   insight,   no   outlook,    apparently   no   sense   of 
being  in  the  presence  of  a  great  matter,  at  once  a  power- 
ful   instrument    and   a    subject    of    first-rate    importance. 
What  I  have  said  of  the  term  "transformation"  might 
be  said  with  equal  truth  of  other  great  terms, — of  Func- 
tion, for  example,  which  is  at  length  happily  winning  its 
way   to    due    recognition   in   elementary   instruction, — of 
Invariant  and  Group,  of  which  I  hope  to  speak  at  a  later 
stage — and  especially  of   Relation,   which,   though  long 
current  in  mathematical  literature  as  a  convenient  term 
used   in   a    sense    semi-scientific    (or   semi-technical)    and 
semi-literary,  is  at  length  coming  to  be  recognized,  owing 
to  recent  work  in  the  logical  foundations  of  mathematics, 
as  denoting  better  than  any  other  term  the  ultimate  tissue 


TRANSFORMATION  155 

of  mathematical  science.  For,  while  this  science  is,  as 
we  have  seen,  composed  of  doctrinal  functions,  these 
forms  are  themselves  woven  of  abstract,  or  formal,  re- 
lations. 

In  this  lecture,  I  purpose  to  deal  with  the  mathemat- 
ical idea  denoted  by  the  term  "transformation."  I  need 
not  say  that  an  hour's  lecture  can  neither  impart  much 
knowledge  of  transformation  theory  nor  give  skill  in  the 
use  of  transformations  as  instruments;  for  the  former 
requires  prolonged  study  and  the  latter  is  the  slow-matur- 
ing fruit  of  practice.  What  I  hope  to  do  in  the  hour  is 
to  make  clear  what  mathematicians  mean  by  a  trans- 
formation and  by  a  law  of  transformation;  to  show  how 
fundamental  and  omnipresent  the  process  of  transforma- 
tion is  in  all  our  thinking;  to  give  an  inkling  of  the  end- 
less number  and  endless  variety  of  existing  transforma- 
tions; to  show  how  transformations  appear  now  as  power- 
ful tools  and  now  as  interesting  themes;  to  disclose  the 
intimate  connection  of  the  notion  of  transformation  with 
that  of  relation  and  that  of  function;  and  briefly  to  indi- 
cate how  the  phenomena  of  transformation  in  the  ab- 
stract static  world  of  mathematics  correspond  to  the 
phenomena  of  change  in  the  concrete  dynamic  world  of 
sense. 

Let  me  begin  as  simply  as  I  can — so  simply,  indeed, 
as  possibly  to  suggest  that  transformation  is  a  trivial 
term,  which  it  is  infinitely  far  from  being.  The  notion  of 
transformation  has  its  root  in  the  power  we  have,  when 
given  any  two  objects  of  thought,  to  associate  either  of 
them  with  the  other.  If  a  and  a'  be  two  such  objects,  we 
can,  in  thought,  associate:  (i)  a  with  a!  {a— >  a')\  or, 
conversely,  (2)  a'  with  a(a<—  a');  or  (3)  each  with  the 
other    (a   * >  a').     If  we  do  (1),  we  say  we  have  trans- 


156  MATHEMATICAL   PHILOSOPHY 

formed  (or  converted)  a  into  a'  and  that  a'  is  the  transform 
of  a;  if  we  do  (2),  we  transform  a'  into  a,  which  is  then 
the  transform  of  a';  if  we  do  (3),  we  transform  each  into 
the  other  and  each  is  the  other's  transform. 

Let  us  take  another  step :  suppose  C  is  a  class  composed 
of  a,  b,  c,  and  that  C  is  a  class  composed  of  a',  b' ,  c' .  We 
may  transform  in  the  manner, — according  to  the  law, — 
shown  in  the  table  (1):  {a  — *  a'),  (b  — >  b'),  (c  — >  c');  or  by 
the  law  (2):  (a -^  a'),  ^  — *  c')i  (c  — >b');  or  by  the  law 
(3):  {a  — >  £'),  (£  — >  a'),  (c— ><:');  or  by  the  law  (4): 
(#  — >  &'),  (b  — » c')>  (c  ~~ *  #')>   or  by  tne  law  (5) :  (a  — >  <:')> 

(i  ->  «o»  fr  -*  &') ;  or  by the  law  (6) :  0  ->  c')>  (*  -»  C), 

{c  — >  a')-  ^n  each  case  we  have,  we  say,  transformed  the 
class  C  into  the  class  C,  and  C,  we  say,  is  the  transform 
of  C.  You  see,  too,  that  we  could  conversely  transform 
C  into  C  in  six  corresponding  ways — by  any  one  of  six 
corresponding  laws,  which  we  could  express  by  tables  as 
above.  You  see,  too,  that  we  could  transform  the  classes 
C  and  C  each  into  the  other,  now  by  one  law,  now  by 

another,    as,   for   example   by  the    law    (7):     (a  < >  a'), 

(b< >V),{c< >c')\   or  by  (8):    (a< >b'),   (b  < >  a'), 

(c  < >  c');  and  so  on.  Notice  that  in  each  of  the  trans- 
formations (1),  .  .  .  ,  (6),  each  element  of  C  has  only  one 
transform  and  that  each  element  of  C  is  a  transform; 
each  of  those  transformations  is  a  one-to-one  transforma- 
tion, and  has  a  direction,  or  sense,  namely,  from  C  to  C; 
not  from  C  to  C;  observe  that  in  (7)  or  (8)  the  transfor- 
mation is  again  one-to-one  but  runs  both  ways.  In  each  of 
the  cases  considered,  one  of  the  classes  is  transformed  into 
the  whole  of  the  other,  not  merely  into  a  part  of  it.  But 
such  need  not  be  so;  we  can  transform  two  or  more 
elements  of  C  into  a  same  one  of  C  or  one  of  C  into  more 
than  one  of  C  and  thus  transform  C  into  a  part,  or  sub- 


TRANSFORMATION  157 

class,  of  C,  as  by  the  law  (9) :  (a  — »  a'),  (b  — >  a'),  (c  — »  &'), 
or  into  the  whole  of  it,  as  by  the  law  (10):  {a  —>  af), 
(b  —>  a'),  (c  —>&')>  (c  —>c')-  You  will  note  that  neither 
(9)  nor  (10)  is  a  one-to-one  transformation.  And  by  a 
little  work,  or  a  little  play,  you  will  readily  discover  that 
the  possibilities  of  transformation  increase  or  decrease 
with  the  sizes  of  the  classes  and  vary  with  other  circum- 
stances such  as  whether  the  classes  are  of  the  same  size 
(containing  the  same  number  of  elements),  whether  the 
classes  have  elements  in  common, — overlap  or  intersect, 
as  we  say, — possibly  coincide,  and  especially  whether  the 
classes  are  finite  (like  those  we  have  considered)  or 
infinite  (like  the  class,  for  example,  of  all  the  integers, 
1,  2,  3,  .  .  .  and  so  on  endlessly). 

Let  us  think  a  bit  about  the  effect  of  such  circum- 
stances. 

Suppose  C  has  more  elements  than  C  has;  it  is  plain 
that  no  one-to-one  transformation  of  C  into  C  can  cover 
the  .vhole  of  C,  but  it  can  be  covered  by  transformations 
that  are  not  one-to-one. 

Suppose  a  of  C  is  the  same  as  a  of  C", — C  and  C 
thus  intersecting;  then  in  the  above  transformations 
(1),  .  .  .  ,  (6),  C  itself  contains  the  transform  of  one  of 
its  own  elements,  a  being  the  transform  of  a  in  (1)  and 
(2),  of  b  in  (3)  and  (5)  and  of  c  in  (4)  and  (6). 

Suppose  C  and  C  coincide,— the  elements  of  either 
being  those  of  the  other, — an  important  special  case  of 
intersection;  then,  you  see,  each  of  the  transformations 
(1),  .  .  .  ,  (8)  converts  (transforms)  the  class  C  into  itself. 

Next  suppose  C  to  be  composed  of  all  the  integers 
I,  2,  3,  ...  fly ...  ;  and  suppose  C  to  be  composed  of  all 
the  even  integers  2,  4,  6  ...  ,  211,  .  .  .  ;  you  notice  that  C 
is  a  party  or  subclass,  of  C;  now  let  us  transform  C  by  the 


158  MATHEMATICAL   PHILOSOPHY 

simple  law  which  requires  us  to  associate,  in  thought,  each 
element  of  C  with  its  double;  symbolically  expressed  the 
law  is:  (i  -»  2),  (2  ->  4),  (3  ->  6),  .  .  .  ,  (n  ->  2n),  .  .  .  ; 
you  observe  that  the  class  C  is  thus  transformed  into  the 
class  C  by  a  one-to-one  transformation;  now,  C  is,  as 
said,  a  part  of  C,  so  that  C  has  been,  you  see,  transformed, 
by  a  one-to-one  transformation,  into  a  £<zr£  of  itself;  and 
that  is  remarkable,  for  it  can  not,  as  you  know,  be  done 
with  just  any  class — it  can  not,  for  example,  be  done  with 
a  class  composed  of  one  thing,  or  two,  or  three,  or  a  dozen, 
or  a  dozen  million.  If  a  class  be  such  that  it  can  be  thus 
transformed  into  some  part  of  itself,  it  is  said  to  be 
infinite — an  infinite  class.  The  concept  denoted  by  the 
term,  "  infinite  class,"  is  one  of  the  most  important  of  our 
modern  mathematical  and  philosophical  concepts.  In 
noticing  that  it  is  defined  by  means  of  transformation, 
you  get  a  glimpse  of  the  latter's  fundamental  importance. 
May  I  here  relate  a  bit  of  relevant  personal  experience? 
Some  years  ago  a  student  of  philosophy  and  I  undertook 
to  read  together  a  pioneer  work  in  the  mathematical 
doctrine  of  infinity — the  Paradoxien  des  Unendlichen  by 
Bernhard  Bolzano,  mathematician,  philosopher  and  theo- 
logian. We  came  to  a  passage  where  Bolzano  shows 
that  the  class  of  points  composing  a  straight  line  segment 
is  an  infinite  class.  He  does  it  very  simply  and  very 
clearly  by  showing,  about  as  follows,  that  the  segment 
O 

_ ., 

Fig.  20. 

can  be  transformed,  in  a  point-to-point  fashion,  into  a 
part  of  itself.  Let  y  =\x\  use  the  equation  as  a  law  of 
transformation  converting  a  point  whose  distance  from 


TRANSFORMATION  159 

0  is  x  into  the  point  whose  distance  is  y,  or  %x;  the  point 
O,  its  distance  being  zero,  is  transformed  into  itself,  for 
the  half  of  zero  is  zero;  the  point  I  is  transformed  into  the 
point  |;  and  the  points  between  0  and  I  have  for  their 
transforms  the  points  between  0  and  |;  nothing,  as  you 
see,  could  be  clearer;  yet  our  plan  of  joint  reading  had 
to  be  here  abandoned,  for  my  fellow  student,  prophet  of 
philosophy,  would  not  follow  Bolzano's  reasoning  and 
remained  invincible  to  the  bitter  end.  Can  you  beat  that  ? 
For  a  simple  example  illustrating  both  the  concept  of 
transformation  and  that  of  infinite  class  geometrically, 
consider  Fig.  21.  We  are  going  to  transform  the  class 
composed  of  the  points  of  segment  AD  into  the  subclass 
composed  of  the  points  of  segment  A"D" .  Let  all  the 
points  of  AD  be  joined  to  P;  any  such  join,  say,  PB, 
contains  a  point,  B'  of  A'D';  associate  B  with  B' ;  in 
this  way  segment  AD  is  transformed,  point  for  point,  into 
segment  A'D';  next  join  Q  to  all  the  points  of  A ' B' ';  any 
such  join,  say,  QB',  contains  a  point,  B"  of  A"D"; 
associate  B'  with  B";  in  this  way  segment  A'D'  is  trans- 
formed, point  for  point,  into  segment  A"D" ;  now  observe 
that,  starting  with  any  point  B  of  AD,  the  first  trans- 
formation gives  us  B'  of  A'D' ,  and  the  second  leads  to 
B"  of  A"D" ;  finally,  associate  each  such  initial  point  B 
with  the  final  point  B";  the  result,  as  you  see,  is  a  point- 
to-point  transformation  of  the  entire  segment  AD  into 
one  of  its  parts,  the  segment  A"D";  and  so  we  see,  in  the 
light  of  the  last  transformation,  that  segment  AD  is  an 
infinite  class  of  points;  the  same  is,  of  course,  true  of  any 
other  segment,  however  short,  for,  in  the  foregoing  argu- 
ment, AD  is  any  segment  you  please.  In  passing,  we  may 
notice  also  that  we  can  choose  Q  so  that  segment  A" D" 
shall  be  any  part  we  please    of   AD,  and  we  have  the 


160  MATHEMATICAL   PHILOSOPHY 

Theorem:  that  any  segment,  however  long,  can  be  converted, 
by  a  point-to-point  transformation,  into  any  one  of  its  parts, 
however  short. 

Before  further  illustrating  the  concept  of  mathematical 
transformation,  let  us  ask  what  we  mean  by  the  law  of 
such  a  transformation.  The  answer  is  pretty  evident. 
It  is  that  the  law  of  a  transformation  is  any  rule,  formula, 
scheme  or  device  which,  given  any  one  of  the  elements  or 


objects  we  are  dealing  with,  determines  its  transform  (or 
transforms,  if  it  have  more  than  one).  It  is,  you  should 
note,  a  psychological  affair,  the  law  being  a  device  for 
guiding  the  transfer  of  attention  from  a  given  object  to  a 
definite  other  object  (or  objects).  Such  a  law  may  be 
variously  expressed:  if  the  class  to  be  transformed  be 
small,  it  is  practicable  to  express  the  law  by  tabulation, 
as  by  the  foregoing  tables  (i),  (2),  .  .  .  ,  (10);  this  is  theo- 
retically possible  for  any  finite  class,  but  is  impracticable 
if  the  class,  though  finite,  be  very  large;  to  express  a  law 


TRANSFORMATION  161 

of  transformation  for  an  infinite  class,  by  tabulation,  is 
not  even  theoretically  possible,  but  in  such  cases  we  must 
use  an  incomplete  table  or  ordinary  speech,  as  in  the 
foregoing  example  where  each  integer  is  associated  with 
(transformed  into)  its  double  and  as  in  the  example  of 
Fig.  21 ;  or,  finally,  we  must  express  the  law,  as  in  Bol- 
zano's example,  by  means  of  an  equation  or  system  of 
equations.  Of  all  methods  of  expression,  the  equational 
method  is  the  most  common,  and  is  usually  the  most 
satisfactory  when  it  is  possible,  which  it  is  sometimes  not. 
We  have  seen  that  the  meaning  of  mathematical 
transformation  has  its  root  in  the  power  we  have  to  asso- 
ciate any  idea  or  thing  with  any  other,  however  like  or 
unlike  the  former.  As  this  power  is  fundamental  and  is 
continually  exercised  by  all  human  beings  in  every  kind  of 
matter,  we  find,  as  we  should  expect  to  find,  not  only 
that  mathematical  transformation  pervades  mathematical 
thinking,  but  that  such  transformation  is  only  a  refinement 
of  a  process  present  in  all  our  human  thinking:  a  fact 
clearly  illustrating  the  general  truth  that  mathematical 
activity,  instead  of  being  remote  from  common  life, 
merely  consists  in  doing,  with  a  peculiar  finesse  and 
ideality,  what  all  human  beings,  when  they  think  about 
the  ordinary  affairs  of  life  and  the  world,  are  doing  in  a 
fashion  relatively  rough  and  crude.  An  ordinary  dic- 
tionary, for  example,  is  a  good  illustration  of  a  kind  of 
transformation  that  would  be  genuinely  mathematical 
were  it  more  precise;  for,  by  definition,  the  class  of  words 
is  transformed  into  the  class  of  verbal  meanings,  and, 
conversely,  the  latter  class  is  transformed  into  the  former; 
the  transformation  runs  both  ways,  but  it  is  not  one-to-one, 
since  a  given  word  commonly  has  two  or  more  meanings 
and  to  a  given  meaning  may  correspond  two  or  more 


162  MATHEMATICAL   PHILOSOPHY 

words  (synonyms).  A  telephone  directory  is  a  similar 
example,  more  nearly  mathematical  than  the  other  one. 
Indeed,  you  will  find,  by  a  little  looking  about,  that  such 
illustrations  abound  on  every  hand.  A  perfect  example  of 
a  genuinely  mathematical  one-to-one  transformation  run- 
ning both  ways  is  afforded  by  the  vulgar  process  of  counting 
the  objects  of  a  class,  a  class  of  count-words  (one,  two,  .  .  .) 
being  transformed,  in  a  certain  order,  into  the  class  (of 
objects  to  be  counted),  and  conversely. 

A  moment  ago  I  told  you  that,  when  a  boy,  my  second 
puny  introduction  to  the  concept  of  mathematical  trans- 
formation occurred  in  a  dry  little  chapter  (called  trans- 
formation of  coordinates)  located  near  the  middle  of  a 
beginners'  course  in  analytical  geometry.  You  will  recall 
that  in  a  previous  lecture  of  the  present  course  I  gave  a 
very  brief  introduction  to  the  analytical  geometry  of  the 
plane.  As  you  will  recall,  it  was  shown  how  to  transform 
(in  one-to-one  fashion)  the  class  of  the  plane's  points  into 
the  class  of  the  real  number  pairs  (x,  y),  and,  conversely, 
the  latter  class  into  the  former;  we  saw  that  the  former 
transformation  gives  birth  to  the  method  of  analytical 
geometry;  and  the  converse  transformation,  to  the  con- 
verse method — that  of  geometric  analysis.  We  may 
think  of  the  two  transformations  as  one  having  two 
converse  aspects,  and,  following  usage,  may  speak  of  the 
two  methods  as  one — called  analytical  geometry.  Now 
observe  carefully  that  the  analytical  geometry  (of  the 
plane),  instead  of  merely  using  transformations  among 
its  processes,  actually  springs  out  of — owes  its  very 
existence  to — a  transformation,  that  of  points  into  num- 
ber pairs  and  of  such  pairs  into  points.  Nay,  the  whole 
of  analytical  geometry  viewed  (properly)  as  a  method  is 
simply  a  vast  transformation  based  upon  the  one  just 


TRANSFORMATION  163 

stated.  To  envisage  the  matter  in  a  large  way,  conceive 
two  immense  canvases  suspended  parallel  to  each  other 
each  of  them  bisecting  the  universe  of  space.  Imagine 
yourself  comfortably  seated  between  them;  fancy  that 
on  the  face  of  one  of  them  are  marked  and  drawn  the 
points  of  a  plane,  its  point  loci, — curves  and  sets  thereof 
limitless  in  number  and  variety, — and  that  on  the  face  of 
the  other  canvas  are  recorded  the  real  number  pairs,  pair 
systems — (#,  y)-equations  and  sets  thereof,  more  numer- 
ous than  the  sands  of  the  sea;  choose  a  unit  of  length 
and  in  the  former  face  a  pair  of  axes.  What  happens? 
You  behold  the  phenomena  on  either  sheet  transformed 
into  those  of  the  other — and  this  infinitely  multitudinous 
transformation  is  the  method  of  plane  analytical  geometry. 
The  like  is  true  of  analytical  geometry  of  three  or  more 
dimensions.  You  see  that  instead  of  transfomation 
being  a  chapter  in  analytical  geometry,  the  latter  is  itself 
only  a  huge  chapter  in  the  infinitely  more  embracing 
theory  of  Transformation. 

And  now  do  you  ask  what  transformations  are  good  for? 
That  is  very  much  like  asking  what  Thinking  is  good  for; 
for  without  transformations,  thinking  could  not  go  on. 
We  have  just  seen  that  analytical  geometry  is  born  of 
transformation  and  does  its  work  thereby;  we  have  seen 
that  the  Olympian  concept  of  Infinity  owes  its  birth  to 
transformation;  we  have  seen  that,  except  for  transforma- 
tion, we  could  not  even  count  the  cattle  in  a  field;  we  have 
seen  that  transformation  pervades  the  practical  thinking 
of  the  workaday  world;  in  previous  lectures,  as  you  will 
recall,  we  saw  that  certain  simple  transformations, — 
the  inversion  transformation  and  others, — enable  us  to 
establish  divers  verifications  of  the  doctrinal  functions  of 
Hilbert,  doctrines  being  thus  derived  from  doctrines  and 


161  MATHEMATICAL    PHILOSOPHY 

compared  with  one  another,  element  for  element  and 
proposition  for  proposition,  giving  rise  in  this  way  to 
what  might  be  appropriately  called  the  Comparative 
Anatomy  of  Doctrines.  These  examples  of  the  use  of 
transformation,  though  they  must  suffice  for  the  present, 
are  only  as  pebbles  picked  up  at  random  on  the  ocean 
beach. 

"  The  hour  contracts  "  and  we  have  yet  to  speak  of 
The  Connection  of  the  Concept  of  Transformation  with 
that  of  Relation  and  that  of  Function. — We  shall  see  that 
the  three  concepts  are  very  similar — but  three  aspects 
indeed  of  one  and  the  same  thing  seen  from  different  points 
of  view.  Let  us,  in  the  first  place,  try  to  understand 
clearly  what  a  relation  is.  This  is  necessary  because, 
though  countless  hosts  of  relations  are  present  everywhere 
in  the  world  and  are  used  by  everybody  all  the  time,  even 
in  their  dreams,  yet  the  scientific  conception  of  what  a 
relation  precisely  is,  is  not  familiar;  even  the  great  major- 
ity of  logicians  and  mathematicians  are  not  familiar  with 
it;  it  seems  a  little  strange  that  such  is  the  fact,  for  the 
logical  theory  of  relations,— the  logical  theory  having  the 
nature  and  the  properties  of  relations  as  subject-matter, — 
goes  back  to  the  logical  work  of  J.  H.  Lambert  (1728-77), 
—mathematician,  physicist,  astronomer, — and  especially 
to  that  of  Augustus  DeMorgan  (1806-78), — mathema- 
tician, logician  and  wit;  was  advanced  by  important 
researches  of  our  fellow  countryman,  the  late  C.  S.  Peirce, 
who  called  it  the  logic  of  relatives;  and  has  now  reached 
a  high  state  of  development  in  the  Principia  of  Whitehead 
and  Russell,  who  have  made  the  theory  of  abstract  rela- 
tions supreme  in  logical  doctrine.  This  great  theory — 
fundamental  alike  in  philosophy  and  in  mathematics — 
has   not   yet   become   in   most   universities   a   subject  of 


TRANSFORMATION  165 

regular  instruction,  but  it  is,  I  believe,  destined  by  its 
intrinsic  importance  to  win  such  recognition. 

Relations,  as  we  are  presently  to  see,  are  determined 
by  propositional  functions  of  two  or  more  variables,  and 
are  accordingly  described  as  dyadic,  triadic  (3-cornered), 
tetradic  (4-cornered),  .  .  .  ,  n-adic  (w-cornered),  and  so 
on.     The  most  important  ones  are  the  dyadic  relations. 

What  is  meant  by  a  dyadic  relation?  I  will  answer 
as  clearly  and  simply  as  I  can  and  will  do  so  by  the  help 
of  two  familiar  examples.  Consider  the  two  propositional 
functions:  (1)  2xJriy  —  1  =0;  (2)  x  is  a  parent  of  y. 
Each  of  these  is  said  to  determine  a  dyadic  relation. 
What  is  the  relation  determined  by  (1)?  We  see  that 
(1)  will  be  satisfied  if,  for  example,  we  replace  x  by  1  and 
y  Dv  —  3  ana<  so  we  saY  tnat  the  ordered  pair  (1,  —  |)  is  a 
couple  of  verifiers  of  (1);  another  such  couple  is  (0,  §); 
there  are,  you  see,  infinitely  many  such  couples;  the  set, 
or  class,  of  all  the  couples  of  verifiers  of  (1)  is  said  to  be 
the  relation  determined  by  (1).  Each  of  the  couples  may 
be  called  an  element  or  constituent  of  the  relation.  What 
is  the  relation  determined  by  (2)  ?  Suppose  John  Smith  is 
the  father  of  Bill  Smith,  then  the  ordered  pair  (John 
Smith,  Bill  Smith)  is  a  couple  of  verifiers  of  (2);  the  class 
of  all  such  verifying  couples  is  the  relation  determined  by 
(2).  In  the  light  of  these  simple  illustrations  you  will 
rightly  understand  that  a  dyadic  relation  is  the  class  of 
all  the  couples  (x,  y)  that  verify  (satisfy)  some  proposi- 
tional function  F(x,  y)  containing  two  (and  only  two) 
variables,  say,  x  and  y. 

It  is  necessary  to  note  carefully  the  following  distinc- 
tion in  usage:  in  ordinary  function-theory — say,  in  algebra 
or  in  analytical  geometry — it  makes  no  essential  difference 
whether  the  #-terms  in  a  propositional  function  come  first 


166  MATHEMATICAL   PHILOSOPHY 

or  the  y-terms  first;  that  is,  for  example,  no  essential 
difference  between  (i)  2#+3;y  —  I  =0,  or  x=\(l  —  3y) 
and  (i')  3y+2^  — 1=0  or  y  =  f(i  —  2x);  (i)  and  (Y).  for 
example,  represent  the  same  straight  line;  but  in  relation- 
theory  it  is  essential  to  take  account  of  the  order  in  which 
the  variables  occur;  the  relations  determined  respectively 
by  (i)  and  (i')  are  not  the  same;  for  example,  the  former 
relation  contains  the  couple  (0,  \)  but  the  latter  does  not; 
on  the  other  hand,  the  latter  contains  the  couple  (|,0  ) 
but  the  former  does  not;  if  we  denote  the  former  relation 
by  R  and  the  latter  one  by  R',  we  then  write  0i?f  to  say 
that  "  0  has  the  relation  R  to  |  "  and  write  f  R'O  to  say  "  £ 
has  the  relation  Rr  to  0  ";  but  we  have  neither  0R'^  nor 
IRO,  for  both  of  these  propositions  are  false;  in  general, 
as  you  see,  if  we  have  xiRyi,  then  we  have  y\R'x\,  and 
conversely,  but  not  yiRxi  nor  x\R'y\  (except  in  very 
special  cases);  here  we  encounter  the  important  notion 
of  the  converse  of  a  relation — two  relations,  R  and  R', 
are  each  the  other's  converse  if  they  are  such  that,  when- 
ever one  of  them  holds  between  the  terms  of  a  couple 
(/i,  fy),  the  other  holds  between  the  terms  of  the  inverse 
couple  (t2,  ti);  thus  the  relations  determined  by  the  propo- 
sitional  functions  (i)  and  (V)  are  each  the  other's  converse. 
It  is  sufficiently  obvious  that  every  relation  has  a  converse. 
(In  the  case  of  some  important  relations, — such  as  equal- 
ity, for  example,  or  similarity  or  diversity  or  identity, — 
the  relation  and  its  converse  are  the  same.)  The  con- 
verse of  the  relation  determined  by  (2)  is  that  determined 
by  the  propositional  function  (2'),  y  is  a  child  of  x,  so  that, 
if  the  couple  (John  Jones,  Mary  Jones)  be  a  constituent  of 
the  former  relation,  the  couple  (Mary  J,  John  J)  is  a  con- 
stituent of  the  latter. 

You  are  now  in  a  fairly  good  position  to  see  that  the 


TRANSFORMATION  167 

concept  of  mathematical  transformation,  the  concept  of 
relation  and  the  (ordinary)  concept  of  mathematical 
function  are,  as  I  have  said,  virtually  but  three  aspects 
of  one  and  the  same  thing  seen  from  different  points  of 
view.  For  look  again  at  the  propositional  functions: 
(i)  #=§(i—  33;),  (i')  y=|(i—  2x).  Observe  that  (i)  is: 
(a)  a  law  of  transformation  by  which  a  class  of  numbers  y 
is  converted  into  a  class  of  numbers  x;  (b)  a  determiner 
of  a  relation,  namely,  that  composed  of  the  couples 
(x,  y)  verifying  (i);  (c)  a  determiner  of  x  as  a  function 
(in  ordinary  sense)  of  y,  namely,  the  function  §(i  —  3y). 
Observe  that  (i')  determines  at  once  the  converse  trans- 
formation, the  converse  relation  and  the  converse, — com- 
monly called  the  inverse, — function.  Observe  that  if 
#1  and  yi  verify  (i),  then  the  pair  {x\,  y\)  is:  (d)  composed 
of  a  thing  transformed  and  its  transform;  (e)  a  pair  of 
values  of  the  function;  (/)  a  couple,  or  constituent,  of  the 
relation.     Of  course,  the  like  is  true  of  (i'). 

Look  again  at  the  propositional  functions:  (2)  x  is  a 
parent  of  y;  (2')  y  is  a  child  of  x.  You  see  that  (2)  is  at 
once:  (g)  the  determiner,  or  law,  of  a  transformation,  asso- 
ciating any  given  x  (a  parent)  with  some  y  or  y's  (child  or 
children  of  the  x),  the  y  or  y's  being  the  x's  transform  or 
transforms;  (h)  the  determiner  of  x  as  a  function  (in 
ordinary  sense)  of  y,  for  to  any  value  of  y  (some  child) 
there  corresponds  a  value  or  values  of  a;  (some  parents); 
(i)  the  determiner  of  a  relation,  composed  of  the  couples 
of  verifiers  of  (2).  It  is  plain  that  (2')  yields  the  respect- 
ive converses  of  the  foregoing  transformation,  function 
and  relation. 

The  connections  shown  by  these  particular  examples 
hold  in  general:  given  a  transformation,  you  have  a 
function  and  a  relation;    given  a  function,  you  have  a 


168  MATHEMATICAL   PHILOSOPHY 

relation  and  a  transformation;  given  a  relation,  you  have 
a  transformation  and  a  function:  one  thing — three  aspects; 
and  the  fact  is  exceedingly  interesting  and  weighty. 
Impressed  by  the  immeasurable  scope  of  the  ordinary 
function  concept,  some  thinkers  have  said,  with  a  striking 
approximation  to  truth,  that  mathematics  and  indeed 
the  whole  of  science  is  just  the  study  of  functions.  It 
can,  you  see,  be  said,  with  the  same  approximation  to 
truth,  that  the  whole  of  science,  including  mathematics, 
consists  in  the  study  of  transformations  or  in  the  study 
of  relations. 

Time  is  lacking  for  extensive  pursuit  of  the  matter 
here.  Before  leaving  it,  however,  I  should  like  to  sig- 
nalize the  parallelism  in  another  way.  A  relation  R 
has  what  is  called  a  domain, — -the  class  of  all  the  terms 
such  that  each  of  them  has  the  relation  to  something  or 
other, — and  also  a  codomain — the  class  of  all  the  terms 
such  that,  given  any  one  of  them,  something  has  the 
relation  to  it;  a  transformation  T  proceeds  from  a  class, — 
that  of  the  things  transformed, — to  a  class — that  of  the 
transforms;  the  independent  variable  (or  argument,  as 
it  is  called)  of  a  junction  F  has  a  range, — the  class  of  values 
the  argument  may  take, — and  the  function  has  a  range, — 
the  class  of  values  the  function  may  take.  Note  the 
matching  of  the  foregoing  things;  it  is  easiest  to  do  it  by 
an  example.  Consider  the  simple  propositional  function: 
x  =  2y.  It  determines  a  relation  R,  a  transformation  T 
and  a  function  F  (i.e.,  x,  or  iy).  Let  K  denote  the  class 
of  real  numbers  and  K'  the  class  of  their  doubles.  You 
see  that  K  is  at  once  the  codomain  of  R,  the  class  trans- 
formed by  T,  and  the  range  of  F's  argument  y;  also  that 
K'  is  at  once  the  domain  of  R,  the  class  of  transforms 
(under  T),  and  the  range  of  F.     In  this  particular  example 


TRANSFORMATION  169 

we  happen  to  have  identity,  or  coincidence,  of  domain 
and  codomain,  of  transform  class  and  transformed  class, 
of  function-range  and  argument-range;  but  this  is,  in  gen- 
eral, not  so;  and  I  recommend  that  you  do  the  matching 
by  some  other  propositional  function,  say,  x  is  the  husband 
of  y,  or  x  is  the  specific  gravity  of  y,  or  the  integer  x  is 
greater  than  the  integer  y,  or  x  is  ethically  so  sublime  that 
he  should  not  allow  y  to  make  a  glass  of  beer  or  "  turn 
water  into  wine." 

I  can  not  refrain  from  tarrying  here  long  enough  to 
illustrate,  by  just  one  example,  the  now  evident  fact  that 
any  problem,  process  or  operation  having  to  do  with 
(ordinary)  functions  is  a  problem,  process  or  operation 
having  to  do  with  relations  or  with  transformations,  and 
conversely.  The  example  is  as  follows:  If  Ri  and  R2 
be  two  relations  such  that  it  is  significant  (true  or  false 
and  not  merely  nonsensical)  to  say  that  Ri's  codomain 
and  i^'s  domain  intersect,  then  there  is  a  relation  R'  — 
called  the  relative  product  of  R\  by  R2- — such  that,  if 
xR\y  and  yR2Z,  then  xR'z\  respecting  functions  the  corre- 
sponding fact  is  this— if  F\  and  F2  be  two  functions  such 
that  it  is  significant  to  say  that  the  range  of  F\s  argument 
intersects  the  range  of  F2,  then  there  is  a  function  F' — 
which  might  be  called  the  functional  product  of  F\  by  F2 — 
such  that,  if  x  =  Fi(y)  and  y=F2(z),  then  x=F'(z); 
finally,  as  to  transformations,  the  corresponding  fact  is 
this — if  T\  and  T2  be  two  transformations  such  that  it  is 
significant  to  say  that  the  class  of  transforms  (under  T\) 
and  the  class  of  things  transformed  by  T2  intersect, 
then  there  is  a  transformation  ^'—called  the  product  of 
T\  by  T2 — such  that,  if  T\  converts  x  into  y  and  T2 
converts  y  into  z,  then  T'  converts  x  into  z.  I  hope  that 
what  I  have  now  said  is  sufficient  to  make  clear  the  exceed- 


170  MATHEMATICAL   PHILOSOPHY 

ingly  important  fact  that  the  meanings  of  the  great  terms 
— transformation,  relation  and  function  (in  ordinary 
sense) — are  essentially  identical. 

The  Rhetoric  of  Mathematics. — Before  closing  this  lec- 
ture I  wish  to  say  something  about  the  psychology  of 
the  mathematician's  use  of  the  word  transformation  and 
in  connection  therewith  to  speak  briefly  of  what  may  be 
called  the  rhetoric  of  mathematics,  a  subject  worthy  of 
much  fuller  treatment  than  we  have  time  to  give  it  here. 
Are  mathematicians  rhetoricians?  Rhetorician?  "That 
is,  of  all  things  " — the  mathematician  will  say — "  exactly 
what  I  most  certainly  am  not."  And  he  should  not  be 
harshly  blamed  for  disowning  the  character;  for,  by 
empty-headed  advocates  of  good  causes  and  by  full- 
headed  advocates  of  bad  ones,  the  art  of  rhetoric  has 
been  so  much  abused  in  the  world  that  "  rhetorician  " 
has  come  to  be,  oftener  than  not,  a  term  of  reproach. 
Nevertheless  Rhetoric  is  a  perfectly  good  name  for  the 
greatest  of  all  the  arts— the  art  of  expression  by  speech. 
"  Thought,"  said  Henri  Poincare,  "  is  only  a  flash  of  light 
between  two  eternities  of  darkness,  but  thought  is  all 
there  is."  How  much  poorer  we  should  be,  had  the  great 
thinker  not  expressed  this  thought,  so  beautiful  and  so 
poignant,  all  will  know  who  have  worthily  meditated  upon 
life  and  the  world.  Thought  unexpressed  is  thought  con- 
cealed, and  concealed  thought — light  hid  under  a  bushel — 
fades  and  perishes  with  the  thinker.  Expressed,  how- 
ever, it  lives  and  grows,  engendering  its  kind,  adding  its 
flame  to  the  flame  of  other  thought,  and  so  that  radiance 
which  is  "  all  there  is  "  increases  and  tends  to  abide:  it  is 
expression,  and  especially  expression  in  speech, — expression 
by  the  art  of  rhetoric, — that  gives  increase  and  perpetuity 
of  light  to  the  narrow  vale  between  the  dark  eternities. 


TRANSFORMATION  171 

And,  now,  rightly  using  the  term  "  rhetoric  "  to 
denote  the  art  of  expression  by  speech,  my  thesis  is  that 
mathematicians  are  all  of  them  devoted  rhetoricians  and 
the  best  of  them  masters  of  the  art.  The  thesis  is  not 
difficult  to  maintain.  For  what  does  the  art  demand  ? 
What  are  the  first  qualities  of  Style?  Clarity?  Energy? 
Order?  Unity?  Convincingness?  Restraint?  Beauty? 
In  respect  to  these  things  no  literature  surpasses  the 
literature  of  mathematics.  It  may  not  indeed  be  easy 
to  understand,  for  the  understanding  of  it  requires  a  fair 
measure  of  mind, — imagination,  especially,  and  logical 
sense, — but  the  difficulty  inheres  in  the  subject  and  not 
in  the  manner  of  handling  it,  for  the  latter  is  clear — 
clear  in  its  definitions,  clear  in  its  enunciations,  clear  in 
its  demonstrations;  its  energy  may  not  be  easy  to  feel, 
for  the  feeling  of  it  requires  a  certain  order  of  sensibility, 
but  energy  is  always  present  in  a  high  degree — indeed 
the  whole  vast  symbolism  of  mathematics,  invented  with 
a  view  to  the  effective  use  of  intellectual  energy,  is  charged 
therewith  beyond  the  measure  of  common  words;  its 
order  may  not  be  easy  to  appreciate,  for  it  is  the  order  of 
logic,  beginning  with  principles  and  pursuing  their 
destined  consequences  under  the  subtle  rule  of  fate;  its 
unity  may  not  be  easy  to  grasp,  for  it  is  the  unity  of  a 
whole  owing  its  integrity  to  the  inner  bond  of  implication; 
its  convincingness  may  not  be  easy  to  sense  for  it  is  disin- 
terested, dispassionate,  purely  intellectual,  ideal;  its 
restraint  is  the  restraint  of  direct  achievement  by  the 
simplest  means;  and  its — Beauty?  Its  beauty  is  two- 
fold: the  exquisite  austere  beauty  of  sheer  form;  and  a 
unique  kind  of  dictional  beauty,  due  to  the  union,  in  mathe- 
matical nomenclature,  of  two  qualities  not  elsewhere 
united.     I  mean  a  certain  literary  quality,  not  essential 


172  MATHEMATICAL   PHILOSOPHY 

to  mathematics  as  such,  and  a  certain  perfection  of  logical 
quality  which  neither  "  the  literature  of  power "  nor 
(outside  of  mathematics)  "  the  literature  of  knowledge  " 
attains.  Pray  do  not  fear  that,  in  saying  this,  I  am 
speaking  as  a  partisan.  Why  should  I?  Mathematics 
and  literature  are,  both  of  them,  ineffably  precious.  I  am 
merely  endeavoring  to  state  an  interesting  fact.  And  if 
the  meaning  and  the  truth  of  what  I  have  said  respecting 
mathematical  diction  be  not  yet  sufficiently  evident  to 
you,  they  will  become  so  if,  when  you  have  the  oppor- 
tunity, you  will  examine  the  matter  attentively.  It 
would  be  sufficient  to  examine  fifty  or  a  hundred  represent- 
ative mathematical  terms,  such,  for  example,  as  the  fol- 
lowing, taken  quite  at  random  from  a  vast  multitude: 
Real  —  ideal  —  imaginary  —  transcendental  —  elliptic  — 
parabolic  —  hyperbolic  —  value  —  range  —  field  —  do- 
main —  harmonic  —  anharmonic  —  symmetric  —  asym- 
metric —  golden  section  —  degrees  of  freedom  —  determi- 
nation— necessary —  sufficient  —  discriminant  —  determi- 
nant —  variable  —  constant  —  invariant  —  covariant  — 
calculus  —  congruent  —  divergent  —  oscillating —  max- 
imal —  minimal  —  sheaf  (of  lines)  —  pencil  (of  planes)  — 
family  (of  curves)  —  cluster  (of  spheres)  —  asympotic  con- 
tact or  approach — point  of  osculation — conjugate  (ele- 
ments or  figures  or  forms) — interval — neighborhood — 
correlation  —  dependent  —  independent  —  closed  —  open 
—  boundary  —  inside  —  outside  —  on  —  slope  —  continu- 
ity —  discreteness  —  finite  —  infinite  —  infinitesimal  — 
limit  —  chance  - —  law.  The  literary  significance  of  such 
representative  terms — the  wealth  and  variety  of  their 
general  meanings,  the  warmth  of  some  of  them,  their 
colors,  the  imageries  awakened  by  them,  the  associations 
they  carry — all  that  is  evident.     In  addition  to  that,  each 


TRANSFORMATION  173 

of  them  denotes,  as  you  may  ascertain,  a  sharply  defined 
mathematical  concept,  which  in  every  instance  is  due  to 
selecting  and  refining  some  feature  or  aspect  of  the  term's 
general  meaning.  We  have,  then,  as  you  see,  in  each  of 
the  terms  two  distinct  qualities — the  literary  quality  of 
its  general  meaning  and  the  logical  quality  of  perfect 
precision  of  specific  meaning;  but  that  is  not  all;  not 
only  are  the  two  qualities  present  in  the  terms,  but  they 
are  connected  in  them — they  are  there  joined  in  a  spiritual 
union  not  to  be  found  beyond  the  borders  of  mathematical 
speech. 

I  had  not  intended  to  speak  at  so  great  length  of 
mathematical  rhetoric  and  can  offer  no  plea  in  mitigation 
except  the  fascination  of  the  theme  and  a  growing  sense  of 
its  importance.  I  must  now  hasten  to  say  in  connection 
therewith,  what  I  have  so  long  delayed  saying  with 
respect  to  the  psychology  of  the  mathematician's  use  of 
the  word  "  transformation." 

Functions,  propositional  functions,  doctrinal  functions, 
propositions,  classes,  points  and  point  configurations, 
numbers  and  systems  thereof — mathematical  entities  in 
general,  simple  or  complex,  elemental  or  composite, — are, 
all  of  them,  stable  things;  immobile  and  immutable;  they 
neither  come  nor  go;  they  are  not  born  and  they  do  not 
perish;  they  have  neither  origin  nor  destiny,  neither  past 
nor  future;  they  are  timeless— inhabitants  of  eternity; 
they  are:  the  world  of  mathematical  entities  is  a  static 
world;  it  owes  its  unity  and  integrity  to  the  presence 
within  it  of  an  infinite  system  of  interlocking  relations; 
and  those  mathematical  relations,  too,  like  the  entities 
constituting  them  and  related  by  them,  are  static.  And, 
now,  what  term  do  mathematicians  employ  to  denote  these 
static  things  ?     They  employ,  as  we  have  seen,  the  dynamic 


174  MATHEMATICAL   PHILOSOPHY 

term  Transformation — as  if  they  fancied  themselves  to  be 
dealing  with  temporal  things,  with  actual  vicissitudes,  with 
transmutations,  with  the  changeful  phenomena  of  the 
fluctuant  world  of  sense.  Why?  Partly,  no  doubt, 
because  they  enjoy  the  illusion,  for  it  stimulates  their 
minds,  enveloping  the  train  of  their  abstract  thought  in 
a  beautiful  mist  of  sensuous  imagery,  and  does  so  without 
diverting  it  from  its  true  course.  But  is  the  illusion 
really  an  illusion?  In  a  sense  it  is — in  the  sense  indicated; 
but  in  a  deeper  sense  it  is  not.  For  in  dealing  with  the 
static  world  of  immobile,  unchanging,  eternal  things, — the 
world  of  concepts, — we  are  in  fact  dealing  with  the 
dynamic  world  of  mobile,  changing,  temporal  things, — 
the  streaming  world  of  sense, — in  the  only  way  in  which 
the  latter  can  be  dealt  with  by  logical  thought.  What  is 
that  way?  Thought  arrests  the  chaotic  stream  and  gives 
it  order — arrests  it,  I  mean,  and  gives  it  order,  in  the 
sense  of  carving  and  shaping  its  confused  and  formless 
content  into  permanent  kinds,  classes,  concepts — unchang- 
ing, immobile  conceptual  entities  constituting  a  static 
world.  For  Thought  this  world  of  static  elements  and 
static  relations  represents,  under  the  aspect  of  eternity, 
the  temporal  elements  and  temporal  transformations  of 
the  dynamic  world  of  Sense.  And  so  we  have  a  kind  of 
provisional  answer  to  our  question:  mathematicians  dis- 
course in  dynamic  terms  about  static  things  because  they 
are  constrained  to  think  in  static  terms  when  they  think 
about  dynamic  things.  The  real  problem,  however,  is  not 
thus  solved — it  is  merely  pointed  at;  for  what  is  the  secret 
of  the  mentioned  constraint  and  the  consequent  compro- 
mise ?  I  can  hardly  state  the  problem  adequately,  much  less 
am  I  able  to  solve  it;  it  is  a  problem  for  you  and  the  future 
— a  momentous  problem  for  science  and  for  philosophy. 


TRANSFORMATION  175 

It  is  evident  that  the  nut  to  be  cracked,  or  one  of  the 
nuts,  is  Time.  We  have  seen  that  in  the  world  of  logic 
things  and  their  relations  are  timeless,  they  are — all  are 
present  at  once;  but  the  things  of  the  other  world  and  their 
transformations  are  temporal,  they  are  not  all  present 
at  once,  but  occur  in  temporal  order — each  thing  becomes 
its  own  successor  and,  in  so  becoming,  ceases  to  be,  so 
that  there  is  a  Past  (which  is  empty)  and  a  Future  (never 
filled) — only  a  mobile  Now,  sole  field  and  vehicle  of  change 
and  transformation.  How  can  either  of  these  sharply 
contrasted  worlds  represent  the  other — the  things  that 
are,  standing  for  the  things  that  happen,  the  permanent 
for  the  changeful,  rest  for  motion,  relations  for  trans- 
formations, the  beginningless  and  everlasting  for  the 
momentary  children  of  birth  and  decay — the  timeless  for 
the  temporal?  It  is  evident,  as  said  before,  that  one  of 
the  troublesome  factors  is  Time.  In  endeavoring  to 
manage  this  factor,  science  has  tried  two  and  only  two 
ways— the  way  of  importation  and  the  way  of  suppression. 
We  are  going  to  see  what  they  are. 

The  former  is  the  way  of  Newton,  the  way  of  his 
Fluxions  and  Fluents.  From  the  objective  dynamic 
world  of  sense  and  physics  time  is  imported  into  the  sub- 
jective static  world  of  conception  and  logic — it  is  smuggled 
in  with  motion:  points  are  not  immobile,  they  move; 
lines  and  curves  do  not  really  exist, — they  are  not  unbe- 
gotten  inhabitants  of  eternity, — they  are  engendered  in 
time  by  motion  of  points;  the  same  is  held  respecting 
surfaces,  which  are  but  the  paths  of  moving  lines  and 
curves;  and  respecting  solids,  produced  by  moving  suf- 
faces;  x,  y  and  z  are  viewed  as  varying  actually,  they 
grow,  their  increments  are  fluents;  and  the  static  world  is 
invaded  by  velocities  and  accelerations.     The  Newtonian 


176  MATHEMATICAL   PHILOSOPHY 

method  of  dealing  with  the  problem, — dynamicising 
the  static  world, — flooding  the  realm  of  eternal  things 
with  the  waters  of  time,— has  had  a  great  vogue,  has 
produced  inestimable  results  and  is  still  dominant;  but 
it  is  not  ultimately  satisfactory;  for  Geftihl  ist  alles,  and 
we  can  not  rid  ourselves  of  the  feeling  that  points  do  not 
move,  that  numbers  do  not  change,  that  relations  are  not 
transmutations,  and  that,  in  general,  logical  and  mathe- 
matical entities  are  immutable. 

And  so,  in  the  recent  literature  of  science  are  to  be 
found  increasing  tokens  of  dissatisfaction  and  reaction. 
The  troublesome  factor  of  Time  is  to  be  suppressed; 
instead  of  dynamicising  the  static  world  of  conception 
and  logic,  we  are  to  staticise  the  dynamic  world  of  sensa- 
tion and  physics.  I  have  alluded  to  tokens.  The  atmos- 
phere of  present-day  "  relativity  "  discussion  is  charged 
with  them.  Let  me  direct  your  attention  to  a  striking 
one.  I  refer  to  Minkowski's  famous  interpretation  of 
what  is  known  as  the  Lorentz  transformation.  My  pres- 
ent concern  is  with  a  single  feature  of  the  interpretation. 
It  may  be  set  in  light  as  follows.  Think  of  a  "  substan- 
tial "  particle  p  of  our  physical  world;  we  are  accustomed 
to  saying  that,  at  a  time-instant  (t),  p  is  at  a  space-point 
(x,  y,  z);  that,  at  instant  (*')>  p  is  at  the  point  (V,  y',  z'); 
and  so  on;  thus,  to  give  account  of  p  we  must  give  both 
its  when  and  its  where — its  t  and  its  x,  y,  and  z;  we  have 
thus  a  tetrad  (x,y,  z,  t);  now  let  us,  says  Minkowski,  view 
the  matter  in  another  way;  let  us  regard  this  tetrad  as 
one  thing  and  name  it  Weltpunkt — world-point;  such  a 
world-point  has  four  coordinates,  x,  y,  z,  t,  and  the 
world  constituted  by  such  points  is  a  4-dimensional 
world;  the  points  of  this  world — of  which  there  are  00  4— 
all  exist  at  once,  they  coexist;   the  fluxion  called  time  is 


TRANSFORMATION  177 

abolished;  motion,  as  a  change  of  place  during  a  flow  of 
time,  is  gone;  in  the  new  world,  where  (if  the  term  be  used) 
has  a  new  meaning — it  has  absorbed  both  the  old  where 
and  the  old  when.  Where  is  the  particle  p\  Where, 
that  is,  in  the  new  world  ?  At  the  point  (x,  y,  z,  t).  Where 
is  the  particle  p'\  At  (#',  yr,  z',  t').  The  particles  p  and 
p'  are  never  the  same;  there  are  relations,  but  no  trans- 
formations; no  history  in  ordinary  sense — no  past — no 
future;  child,  youth,  man  coexist  as  phases  of  one  individ- 
ual; the  same  is  true  of  morning,  noon,  night  and  so  on: 
all  is  static — as  a  "  painted  ship  on  a  painted  ocean." 

You  see  what  has  happened  here  and  how.  By  sup- 
pressing the  fluxional  character  of  time  along  with  its 
implicates, — motion,  transformation,  change, — and  by 
regarding  time  as  simply  a  cosmic  dimension  to  be  joined 
with  the  familiar  dimensions  of  space,  the  Dynamics  of 
our  spatially  3-dimensional  world  has  been  made  to 
appear  as  a  Statics  of  a  4-dimensional  world.  I  need  not 
say  that  this  way  of  handling  time,  however  beautiful 
and  helpful,  is,  like  Newton's  way  (of  which  it  is  the 
antithesis),  not  ultimately  satisfactory.  I  should  add  that 
Minkowski  was  far  from  regarding  it  as  a  final  solution. 

And  so  science  and  philosophy  are  still  confronted  and 
to-day  confronted  afresh  by  the  age-old  problem  of  Time. 
No  one  has  been  able  to  tell  satisfactorily  what  is  meant, 
or  should  be  meant,  by  when.  From  time  immemorial, 
human  beings  have  talked  of  "  instants,"  but  no  one  has 
discovered  what  an  instant  is.  It  is  important  to  observe 
that  the  time  problem  is  not  solitary;  it  is  but  one  of  a 
class  of  kindred  problems  or  is  perhaps  an  aspect  or  a 
fragment  of  a  larger  problem  embracing  them  all.  For 
what  is  meant,  or  should  be  meant,  by  where?  By  her  el 
By  there}     By  a  point?    And  we  talk  of  matter  as  of  time 


178  MATHEMATICAL   PHILOSOPHY 

and  space.  But  what  is  an  atom  or  an  electron?  And 
what  is  ether?  No  one  has  been  able  to  answer  these  and 
like  questions  satisfactorily.  It  is  fair  to  ask  what  sort 
of  answer  would  be  satisfactory.  By  help  of  an  analogy 
this  question  can,  it  appears,  be  answered  approximately 
with  a  good  deal  of  confidence.  Respecting  the  nature  of 
Number  there  are  questions  analogous  to  the  foregoing 
questions  respecting  Time,  Space,  Matter.  As  we  ask 
for  the  meanings  of — instant — when — now — then — point 
—  where  —  here  —  there  —  electron  —  atom  —  ether  — 
and  the  like,  so  in  the  domain  of  Number  we  ask  such 
questions  as  these:  What  is  a  cardinal  number?  A  posi- 
tive integer?  A  negative  integer?  A  rational  number? 
An  irrational  number?  A  real  number?  A  complex 
number?  And  so  on.  Now,  in  recent  years,  by  workers 
in  the  logical  foundations  of  mathematics,  especially  by 
the  researches  of  Peano,  Frege,  Russell  and  Whitehead, 
these  latter  questions — analogues  of  the  former  kind — 
have  at  length  been  answered  with  a  pretty  high  degree 
of  satisfaction.  Answered  how?  Answered  in  terms  of  a 
small  number  of  logical  data  (or  concepts  or  constants) 
more  fundamental  and  more  embracing  than  the  terms 
defined:  answered,  that  is,  in  terms  of  such  familiar 
logical  notions  as  class,  relation,  symmetric  relation,  asym- 
metric relation,  serial  relation,  and  a  few  other  varieties. 
In  this  procedure  in  the  Number  field  we  have  probably 
a  model  of  what  to  seek  in  the  other  fields  and  a  clue  to  it: 
the  ultimate  constituents  of  time,  space  and  matter  are 
to  be  conceived  in  terms  of  logical  data.  When  this  great 
task  is  accomplished,  will  the  results  be  entirely  satis- 
factory? I  suspect  not;  the  problem  of  defining  the 
various  kinds  of  number  in  terms  of  logical  constants  has, 
as  I  have  said,  been  pretty  satisfactorily  solved;    but, 


TRANSFORMATION  179  \ 

unless  I  am  mistaken,  there  remains  a  psychological  prob- 
lem,— an  immense  and  difficult  one, — the  task  of  discov- 
ering the  connections  of  the  number  concepts  with  the 
data  of  sensation  and  sense-perception.  Is  a  class,  for 
example,  or  a  relation,  a  percept  or  a  concept  or  both? 
So,  too,  respecting  time,  space  and  matter,  if  the  problem 
of  defining  their  elements  in  logical  terms  were  solved, 
there  might  still  remain  to  be  solved  a  psychological  prob- 
lem; or  it  may  be  that  the  solutions  of  the  two  problems 
will  be  advanced  simultaneously.  Duration,  for  example, 
seems  to  be  a  datum  of  sense,  and  so,  too,  as  William 
James  long  ago  pointed  out,  voluminousness,  or  bulk, 
appears  to  be  a  datum  of  sense;  it  may  be  that  an  instant 
and  time  itself  will  be  logically  and  psychologically  defined 
in  terms  of  sense-given  durations;  and  that  a  point  and 
space  itself  will  reach  similar  definition  in  terms  of  sense- 
given  bulks.     And  similarly  for  similar  things. 

You  are  to  be  congratulated  on  the  date  of  your 
generation  when  these  kindred  problems,  or  these  kindred 
phases  of  the  one  great  problem  constituted  by  them, 
are  pressing  for  solution;  for  the  problem  is  indeed 
immense,  embracing,  not  merely  the  now  exciting  question 
of  "  relativity,"  but — what  is  infinitely  more — the  nature 
/of  the  ultimate  data  and  ultimate  structure  of  Knowledge. 
Let  me,  in  closing,  refer  you  to  some  of  the  works  of  some 
of  the  pioneers — to  Russell's  Scientific  Method  in  Philoso- 
phy,1 to  his  Analysis  of  Mind,  to  certain  parts  of  White- 
head's Organization  of  Knowledge,2  to  his  Concept  of 
Nature,  and  especially  to  his  truly  momentous  book, 
The  Principles  of  Natural  Knowledge.     Regarding  the  last 

'Reviewed  by  C.  J.  Keyser  in   The  Bulletin  of  the  American  Mathe- 
matical Society. 

3  Reviewed  by  C.  J.  Keyser  in  Science. 


180  MATHEMATICAL   PHILOSOPHY 

it  must  be  said  that  it  contains  obscure  passages.  The 
obscurity  is  to  be  ascribed  partly  to  the  very  great  dif- 
ficulties of  the  subject  and  partly  to  the  new  ideas  throng- 
ing the  author's  mind  and  impatiently  pressing  for  utter- 
ance. The  ideas  will  gradually  win  their  way  to  greater 
clarity  of  exposition  by  Whitehead  himself  or  by  his 
collaborators  and  successors  in  the  work;  for  his  book 
both  makes  and  marks  the  beginning  of  an  epoch,  and, 
when  it  perishes,  it  will  "  perish  by  supersession."  In 
the  same  connection,  you  should  examine  Professor 
Eddington's  Space,  Time,  and  Gravitation. 


LECTURE  XI 
Invariance 

THE   AGES-OLD    PROBLEM    OF    PERMANENCE   AND   CHANGE 

THE  QUEST  OF  WHAT  ABIDES   IN  A  FLUCTUANT  WORLD 

THE  BINDING  THREAD  OF  HUMAN  HISTORY THE  TIE 

OF    COMRADESHIP    AMONG    THE    ENTERPRISES    OF    THE 

HUMAN      SPIRIT NEED      OF      CRITICAL      HISTORY      OF 

THOUGHT. 

Invariance  is,  in  all  strictness,  a  subject  of  universal 
interest:  it  penetrates,  as  we  shall  see,  not  only  all  the 
sciences  and  all  the  arts,  but  also  the  common  life  of 
mankind  everywhere  and  always.  And  no  wonder.  For 
the  most  obvious,  the  most  embracing,  the  most  poignant 
and  the  most  tragic  fact  in  the  pageant  we  call  the  world 
is  the  fact  of  Change;  in  the  world  of  sights  and  sounds, 
in  the  world  of  sense,  nothing  abides.  "  The  life  of  man," 
said  the  Spirit  of  the  Ocean,  "  passes  by  like  a  galloping 
horse,  changing  at  every  turn,  at  every  hour."  And  so 
the  sovereign  fact  in  the  life  of  reason  is  the  quest  of 
things  Eternal.  The  mathematical  theory  of  transforma- 
tions,— dealt  with  in  the  preceding  lecture, — is  the  logic 
of  change;  the  mathematical  theory  of  invariance, — the 
principal  theme  of  the  present  hour, — is  the  logic  of  eternal 
things,  the  logic  of  permanence.  The  latter  theory,  like 
the  former,  with  which  it  has  the  closest  connections,  is 
immense,    manifold,    technical    and    intricate;     extensive 

181 


182  MATHEMATICAL   PHILOSOPHY 

knowledge  of  it  can  be  gained  only  by  pursuing  it  through 
many  months  with  the  tireless  energy  of  a  sleuth-hound. 
It  is  my  aim  to  give  you  just  a  little  introduction  to  the 
matter,  a  clue  to  it,  a  good  grasp  of  its  central  idea,  a  very 
slight  acquaintance  with  its  methods,  and  a  fair  sense  of 
its  general  significance  and  its  bearings  as  a  prototype 
for  that  quest  of  abiding  reality  which  has  dominated  all 
the  great  truth-seeking  activities  of  man  and  has  served 
to  unite  them, — religion,  philosophy,  art,  science, — as  but 
different  aspects  of  one  supreme  enterprise:  emancipation 
from  the  tyranny  of  change — discovery  of  a  stable  world — 
a  haven  of  refuge  from  the  raging  tempests  of  the  sea. 

Let  us  begin  as  we  began  in  the  case  of  transformation 
— as  simply  as  we  can;  indeed,  to  begin  aright  we  must 
return  and  begin  our  new  study  just  where  we  began  to 
study  the  meaning  of  mathematical  transformation;  for 
we  may  say  at  once  that,  in  general  sense,  an  invariant, 
as  the  word  indicates,  is  to  signify  something  which,  when 
other  things  connected  with  it  suffer  change,  remains 
itself  unchanged;  and  now  change,  as  we  have  seen,  is 
represented  in  logic  (in  mathematics)  by  means  of  relations 
which,  as  we  have  also  seen,  mathematicians  call  trans- 
formations; so  that  the  mathematical  term  "  invariant  " 
or  "  invariance  "  would  be  unintelligible  or  meaningless 
save  for  its  connection  with  the  mathematical  notion  of 
transformation. 

We  will  accordingly  suppose,  as  in  the  preceding  lec- 
ture, that  a  and  a'  denote  two  objects  of  thought  and  that 
by  a  transformation — which  we  may  denote  by  T — a  has 
been  transformed,  or  converted,  into  a'y  {a  — >  a').  Now, 
an  object  of  thought  has  what  we  call  properties,  some  (at 
least  one)  of  which  are  peculiar  to  it  and  some  belong  to 
one  or  more  other  objects  as  well.     Let  us  suppose  that  a 


INVARIANCE  183 

has  the  peculiar  properties  p\,  p2, .  •  .  ;  that  a!  has  the 
peculiar  properties  pi,  p2,  .  .  .  ',  and  that  the  properties 
iri,  7T2,  .  .  .  belong  to  both  a  and  a'',  we  say  that  the  proper- 
ties 7ri,  7T2,  .  .  .,  since  they  belong  both  to  a  and  to  its 
transform  a' ,  are  invariant  under  T, — have  suffered  no 
change, — are  preserved;  and  that  pi,  p2y  .  .  •  ,  as  they 
belong  to  a  but  not  to  its  transform,  are  variant  under  T, — 
properties  lost  under  the  transformation, — not  carried 
over  by  it.  It  is  plain  that  under  the  converse  (commonly 
called  the  inverse)  transformation  V ,  (a  <—  a'),  pi,  p2f, . . . 
are  variant  while  in,  7T2,  .  .  .  are  invariant  as  before.  It  is 
evident  that,  if  a  property  be  invariant  under  some 
transformation,  it  will  be  invariant  under  the  converse 
transformation.  I  am  aware  that  what  I  have  now  said 
is  so  general,  abstract  and  simple  as  to  make  the  concept 
dealt  with  seem  unreal — tasteless,  pallid,  thin,  intangible. 
But  the  seeming  is  seeming  only.  The  idea  in  question, 
far  from  being  detached  from  reality,  literally  pervades  it 
— pervades  our  thinking  about  it  and  our  handling  of  it. 
How  may  we  convince  ourselves  that  this  is  true?  We 
may  do  it  by  looking  about  us  a  little  and  by  a  little  reflec- 
tion— by  considering  a  few  specific  concrete  examples 
and  observing  that  such  examples  abound  in  countless 
multitudes  on  every  hand. 

For  a  few  examples  that  everyone  can  understand, 
consider  the  following.  Let  /  denote  the  class  of  familiar 
integers:  I,  2,  3,  .  .  .  ;  and  suppose  these  to  be  trans- 
formed in  accordance  with  the  law:  (1)  y  =  2x.  The  trans- 
forms constitute  the  class  of  even  integers:  2,  4,  6,  ...  . 
We  note  that  integers  are  converted  into  integers,  and  so 
the  property  of  being  an  integer  is  preserved — it  is  an 
invariant  under  transformation  (1);  the  value  of  an 
integer,    however,    is    a    property    not    preserved — it    is 


184  MATHEMATICAL   PHILOSOPHY 

doubled;  neither  is  the  product  of  integers  preserved,  for, 

if  #1  and  #2  be  any  two  numbers  in  /,  and  their  transforms 

be  yi  and  ;y>2,  then  the  product  ;via;2  is  transformed  into 

yi}>2>  which  is  not  #1*2  but  is  4JV1X2;   sums  and  differences 

are   also  variant,  the  transforms  of  #i+#2   and  #i— #2 

being  respectively  2(^1+^2)    and   2(^1—^2);    but  ratios 

yi     2xt     xi 
are  invariant  tor  —  =  —  =  — .     JNow  suppose  the  law  or 

y2     1x2     X2 

transformation  to  be:  (2)  y—\x\  then  neither  the  prop- 
erty of  being  an  integer,  nor  value,  nor  product  nor  sum 
nor  difference  is  invariant,  but  ratio  is;  so,  too,  is  the 
property  of  being  a  number,  as  was  also  the  case  under  (1), 
though  not  there  mentioned.  This  latter  property  is 
again  invariant  under  the  transformation:  (3)  y=x  +  i; 
but  ratio  is  not.  Is  the  property  of  being  a  number  invari- 
ant under  every  transformation  of  P.  No,  it  is  not 
invariant  under  the  transformation  converting  the  inte- 
gers of  /  into  an  endless  succession  of  days,  di,  d.2  ds,  .  .  . 
in  accord  with  the  law:  (4)  n  — »  dn;  but  even  under 
(4)  we  have  an  interesting  invariant, — namely,  the  prop- 
erty of  nextness, — for  the  transform  of  an  integer  next 
after  a  given  one  is  itself  next  after  the  given  one's  trans- 
form. Note  that  under  none  of  the  transformations 
(1),  (2),  (3)  is  the  class  /  invariant  as  a  whole,  being 
converted  by  (1)  and  (3)  into  a  part  of  itself  and  by  (2) 
into  a  class  including  it  and  another  class  besides — a  class 
of  fractions.  But  /  as  a  whole  is  invariant  under  many 
transformations — for  example,  under  the  reciprocal  one- 
to-one  transformation:    (5)  1  < ►  2,  3  < >  4,  .  .  .  . 

For  a  different  sort  of  example,  consider  the  following. 
Let  D  denote  something  very  strong  and  solid,  say  a 
diamond  at  a  certain  time  and  place;  suppose  it  removed 
gently  to  another  place,  and  now  denote  it  by  D';  D  and  D' 


INVARIANCE  185 

are  two  objects  of  thought,  for  they  are  evidently  not  the 
same  in  all  respects.  Let  us  now  suppose  D  transformed 
into  D'y  (D  — »  D'),  by  a  transformation  T—l  do  not  mean 
transmutation,  I  mean  association  of  D  with  Df,  for  that, 
as  we  have  seen,  is  at  bottom  what  a  mathematical 
transformation  is,  that  and  nothing  more.  Among  the 
variant  properties  of  D  under  T  are  certainly  time  and 
place,  and  possibly  weight  and  distance  from  the  Moon. 
What,  if  any,  are  the  invariants?  Subject  to  some  correc- 
tion by  the  refinements  of  modern  physics,  it  is  yet  instruc- 
tive to  answer  that  among  the  invariant  properties  of 
D  under  T  are  shape,  size,  mass,  degree  of  hardness, 
capacity  for  light  absorption,  and  so  on.  Some  of  these 
will,  of  course,  not  be  invariant  under  transformation  of 
D  into  D',  where  D'  denotes  D  crushed. 

For  an  example  drawn  from  a  very  different  field,  let 
P  denote  the  personality  of  John  Smith  at  the  age  of  15, 
P'  his  personality  at  the  age  of  30,  and  let  someone,  say 
Smith  himself  at  the  age  of  45,  transform  P  into  P\ 
(P  — »  P'),  by  a  transformation  T — again  I  do  not  mean 
transmutation,  the  mysterious  process  of  a  boy's  becom- 
ing a  man.  The  variant  properties  of  P  under  T  are 
obviously  many — years,  for  example,  wisdom,  folly,  inter- 
ests, hope,  and  so  on;  among  invariants  are  the  properties 
of  being  a  son,  of  being  a  man,  of  being  a  human,  of  being 
what  Count  Korzybski  calls  a  time-binder,  of  being  a 
visible  object;  another  one — of  extraordinary  interest — 
is  the  property  called  personal  identity.  This  last  prop- 
erty, which  runs  through  a  long  sequence  of  personalities, 
exemplifies  an  immense  class  of  important  invariants  that 
no  one  has  been  able  to  formulate  precisely  though  their 
existence  is  manifest:  we  may  call  them  unformulated 
or  qualitative  invariants.     These  are  not  indeed  strictly 


186  MATHEMATICAL   PHILOSOPHY 

mathematical  invariants,  which  are  formulated  precisely; 
yet  they  evidently  belong,  like  the  latter,  to  the  type  of 
invariantive  matter  and  will  more  and  more  approximate 
or  even  attain  precision  in  course  of  the  progress  of 
analysis  and  definition. 

Let  us  now  have  another  little  dip  in  the  boundless  sea 
of  strictly  mathematical  invariants.  In  Lecture  IV  we 
were  introduced  to  the  -pole-polar  transformation  of  a 
plane  with  respect  to  a  circle;  we  saw  that  it  converts  a 
point  (as  pole)  into  a  line  (as  polar),  and  a  line  (as  polar) 
into  a  point  (as  pole) ;  it  is,  you  see,  a  one-to-one  reciprocal 
transformation — a  twofold  affair  composed  of  a  transform- 
ation (i)  of  points  into  lines  and  the  converse  transforma- 
tion (2)  of  lines  into  points.  Let  us  first  think  of  (1)  alone; 
we  readily  detect  certain  variants  and  certain  invariants 
under  it;  the  property  of  being  a  point  is  not  preserved, 
since  the  transform  of  a  point  is  a  line;  a  range  of  points 
loses  the  range  property,  since  the  transform  of  a  range  is 
a  pencil  (of  lines);  distance  is  lost,  since  the  distance 
between  two  points  has  for  transform  the  angle  between 
two  lines  (the  transforms  of  the  points);  now,  as  you 
know,  a  curve  has  two  aspects  (called  dual  aspects),  one 
as  the  locus  of  its  points,  the  other  as  the  envelope  of  its 
(tangent)  lines;  the  property  of  being  a  curve  is  invariant 
(preserved),  for  under  (1)  the  transform  of  a  curve  is  a 
curve;  but  the  locus  aspect  is  lost,  its  transform  being 
the  envelope  aspect;  as  we  saw  in  Lecture  IV,  the  relation 
of  order  and  the  relation  of  congruence  are  exceedingly 
important  invariants  under  (1);  by  (1)  the  ordinary 
geometry  D\  of  the  plane  was  transformed  into  the 
geometry  D2  of  lines  and  pathopencils;  and,  as  D\  and  D2 
are  the  same  in  respect  to  form,  we  see  that  under  (1) 
doctrinal    form,    or    logical    structure,    is    invariant.      I 


INVARIANCE  187 

leave  to  you  the  analogous  consideration  of  transforma- 
tion (2). 

Let  me  suggest  that,  for  a  handsome  illustration  of 
invariance,  you  look  again  at  the  inversion  transformation, 
presented  and  employed  in  Lecture  V.  You  will  readily 
see  that  the  inversion  plane,  as  a  whole,  and  the  inversion 
circle,  are  invariants;  that  a  point's  property  of  being  a 
point  and  a  curve's  property  of  being  a  curve,  as  well  as 
its  locus  aspect,  are  invariants;  that,  on  the  other  hand, 
neither  the  magnitude  of  distances  nor  the  sense  of  angles 
is  invariant,  but  that  angular  magnitudes  are  preserved; 
and, — most  beautiful  of  all, — if,  as  is  customary,  we  regard 
a  straight  line  as  a  circle  (of  infinite  radius),  then  cocircu- 
larity  of  points  is  an  invariant,  for,  as  we  saw,  the  inverse, 
or  transform,  of  a  circle  is  a  circle. 

I  hope  I  shall  not  be  overtaxing  your  interest  if,  to  the 
foregoing  list  of  somewhat  random  illustrations,  I  add  a 
specially  chosen  one,  lying  at  the  heart  of  Projective 
Geometry.  Let  x  denote  a  real  number  and  suppose  it 
transformed  into  x'  in  accordance  with  the  law: 

,     ax+b 
cx-\-dy 

note  that  (1)  contains  three  parameters, — the  three  inde- 
pendent ratios  of  the  coefficients, — say,  a/d,  b/dy  c/d.  To 
each  of  these  we  may  give  any  one  of  the  infinitely  many 
(00) real  values;  and  thus,  as  you  see,  there  are  00 3  different 
transformations  of  the  form  (1);  the  transform  x'  of  a 
given  x  will,  of  course,  depend  upon  the  particular  one  of 
the  transformations  we  choose  to  employ.  Let  us  now 
imagine  that  (1)  is  some  definite  one  of  the  transforma- 
tions; then  a  given  x  has  a  definite  transform  x'\  let 
#1,   X2,   *3,    #4    be    any   4   given    real    numbers    and    let 


188  MATHEMATICAL   PHILOSOPHY 

X\'y  xo',  xz>  x±  be  their  respective  transforms.  Consider 
the  expression,  or  function, 

(Xi  —X2)  (X3—  #4) 
(X2—X2i)(x4—Xi)' 

being  very  important,  this  function  has  a  special  name; 
it  is  called  the  anharmonic  ratio  of  x\,  X2,  #3,  Xa  taken  in  the 
order  as  written,  and  may  be  denoted  by  the  symbol 
R{x\X2Xzx±)\   so  we  may  write 

n/  N        (#1  ~X2)(X3—  Xa) 

K(XiX2X3Xi)  = ■ — 

{X2—Xz)\X±—X\) 

Now,  the  transform  of  R{x\X2XzXa)  obviously  is 
R(x\X2X2,'xa).  How  are  these  two  anharmonic  ratios 
related?  To  find  the  answer  it  is  sufficient  to  replace 
xi,  X2  y  xz,  X4!  of  the  transform  ratio  by  their  respective 
values 

ax\-\-b      ax2-\-b      axz-\-b      ax4-\-b 
cx\-\-<£     cx2-\-d>      cxz+d''      cx±-\-<£ 

and  to  simplify  the  result;  by  doing  so,  which  is  easy,  you 
will  find  that  R{x\X2X^x\)  =R(xi'xo'x3'xi).  We  have  here, 
as  you  see,  an  exceedingly  beautiful  specimen  of  mathe- 
matical invariance:  namely,  under  each  and  all  of  the 
threefold  infinity  of  transformations  of  form  (1),  each  of 
them  ronverting  the  entire  class  of  real  numbers  into 
itself,  the  anharmonic  ratio,  (R(xiX2XzXa),  of  any  ordered 
set  of  four  numbers,  remains  absolutely  unchanged. 

The  invariant  in  question,  though  it  here  appears  as 
a  function  of  pure  numbers,  lies,  as  I  have  intimated,  at 
the  heart  of  Projective  Geometry.  We  may  see  the 
thing  in  geometric  light  readily  as  follows.  Suppose  Fig. 
22  to  be  in  a  projective  plane;   let  x  be  the  distance  from 


INVARIANCE 


189 


0  to  P;  a  value  of  x  is  thus  associated  with  a  point  P  of 
the  range  L,  with  a  line  p  of  the  pencil  V,  and  conversely; 
R(xiXoX3Xi)  may  be  called  the  anharmonic  ratio  of  the  4 
corresponding  points.  Pi,  P2,  P3>  P4,  and  be  denoted  by 
R(PiP2PzP4),  or  of  the  corresponding  lines,  pi,  po,  pz,  p4, 
and  be  denoted  by  R(pip2p3p4)',  then,  you  see,  any 
tetrad  of  collinear  points  or  any  tetrad  of  copunctal  lines 
has  an  anharmonic  ratio. 


Fig.  22. 

Consider  a  transformation  of  form  (1)  in  connection 
with  Fig.  23;  let  us  associate  the  values  of  x  with  the 
points  of  L  and  the  lines  of  V,  and  the  corresponding 
values  of  x'  with  the  points  of  L'  and  the  lines  of  V ';  we 
have,  you  see,  thus  transformed  the  points  of  the  range 
L  into  the  points  of  the  range  L'  and  also  into  the  lines 
of  the  pencil  V ';  at  the  same  time  we  have  transformed 
the  pencil  V  into  the  pencil  V  and  also  into  the  range  V . 
Now  let  Pi,  P2,  P3,  P4  (or  pi,  p2,  p3,  p4)  be  any  4  points 
(or  lines)  of  L  (or  V),  and  let  their  transforms  be 
Pi',  P>\  P/,  Pi'  (or  px\  p-/,  p/,  p,f)  of  V  (or  PO,  then, 
owing  to  the  invariance  of  anharmonic  ratios  under  our 
transformation,  we  have 


190 


MATHEMATICAL   PHILOSOPHY 


R(PlP2P*Pi)  =*R(pip2p3p*)  = 

i?(iYP27YP4')  =R(pi,p2,PsW) 

Such  a  transformation  is  called  a  projective  transforma- 
tion and,  when  it  has  been  applied  as  above,  L  and  Z/,  or 


Fig.  23. 

V  and  V,  or  L  and  V,  or  //  and  V,  are  said  to  be  pro- 
jectly  related.  Why  call  the  transformation  projective? 
Because  the  correspondence  set  up  by  it  can  be  set  up 
by  what  architects  call  projection — as  is  shown  for  the 
case  of  L  and  V  in  Fig.  24,  where  Pi,  P2,  P3,  •  •  •  are 
projected   from  V  respectively  into  Pi',  B,  J,  .  .  .  ,   and 


INVARIANCE 


191 


these  are  then  projected  from  V  respectively  into 
Pi',  P2',  iY,  .  .  .  ,  so  that  we  have  finally  Pi,  P2,  P3,  .  .  .  , 
corresponding  respectively  to  Pi',  P2',  P3',  .  .  .  ,  as  re- 
quired by  the  given  transformation. 

The  00 3  projective  transformations  of  form  (1)  are 
included  in  a  yet  larger  class  of  projective  transformations 
of  the  plane,  which  latter  are  included  in  a  still  larger 


Fig.  24. 

class  of  projective  transformations  of  (ordinary)  space, 
and  so  on  for  spaces  of  higher  dimensionality.  Imagine 
two  planes  it  and  ir'  in  ordinary  space;  let  F  be  some 
figure  in  r\  let  0  be  a  point  in  neither  plane;  the  lines 
through  0  and  the  points  of  F  project  (as  we  say)  the 
latter  points  into  points  of  71-'  constituting  a  figure  F'. 
F  and  F'  have  certain  properties  in  common;  that  is,  cer- 
tain properties  of  a  figure  are  invariant  under  projection. 


192  MATHEMATICAL    PHILOSOPHY 

Projective  Geometry  is  the  study  of  such  properties; 
these  are  all  of  them  expressible  in  terms  of  anharmonic 
ratios  and  that  is  why  I  said  that  the  invariance  of  the 
anharmonic  ratio  is  at  the  heart  of  projective  geometry. 

The  idea  of  invariance — of  permanence  in  the  midst 
of  change, — of  abiding  realities  in  a  fluctuant  world, — is 
very,  very  old, — far  older  than  history, — as  old  probably 
as  the  race  of  man— certainly  as  old  as  the  dream  of 
eternal  things,  of  everlasting  goods.  On  this  account  and 
especially  because  mathematics  has  always  been  peculiarly 
concerned  with  eternal  things,  it  seems  a  bit  strange  that 
the  mathematical  theory  of  invariance— the  doctrine,  I 
mean,  having  invariance  consciously  for  its  subject-matter 
— is  a  strictly  modern  theory.  Yet  such  is  the  case.  Why 
it  is  so  is  a  question  I  shall  not  here  attempt  to  answer. 
It  is  but  a  minor  one  of  a  large  class  of  very  interesting 
questions  belonging  to  a  great  unwritten  history — the 
history  of  the  development  of  intellectual  Curiosity, — 
a  subject  requiring  for  its  treatment  philosophical  genius 
and  learning  of  the  highest  order.  The  mathematical 
theory  of  invariance  is  about  as  old  as  American  inde- 
pendence. Like  most  other  great  doctrines,  it  began, 
not  in  ratiocination,  but  in  an  observation,  and  not  in  an 
observation  of  a  great  fact  by  a  small  mind  but  in  an 
observation  of  a  small  fact  by  a  great  mind.  I  allude 
to  the  observation  by  Lagrange  in  1773  of  the  little  fact 
that  the  discriminant  of  the  quadratic  expression,  or 
form,  (1)  ax2-\-2bxy-\-cy2,  remains  unchanged  when  (1)  is 
transformed  by  replacing  x  by  x-\-\y.  In  high  school  or 
college,  you  learned  what  the  discriminant  of  (1)  is  and 
what  it  signifies.  May  I  remind  you?  It  is  b2—ac;  and 
it  means  that  the  roots  of  the  equation  (2)  ax2 -\-ibxy -\- 
cy2  =0, — the  two  values  of  x/y  that  satisfy  (2) — are  equal, 


INVARIANCE  193 

or  real  and  unequal,  or  conjugate  imaginary,  numbers 
according  as  b2  —  ac  is  equal  to,  or  greater  than,  or  less 
than  zero.  Transforming  (i)  by  replacing  its  x  by  x :  +  Xy, 
you  will  readily  find  that  the  transform  of  (i)  is 
(V)  ax2+2(a\+b)xy-\-{a\2+2b\+c)y2>  and  that  the 
transform  of  (2)  is  the  equation  (2')  ax2  +2(a\-\-b)xy  + 
{a\2+2b\-{-c)y  =0.  The  discriminant  of  the  transform 
(i')  or  (2')  is  (a\-\-b)2—  a(a\2-\-2b\Jrc),  and  this,  you  see, 
reduces  to  b2—ac  exactly.  You  will  notice  that  by  allow- 
ing X  to  vary  in  value,  we  obtain  an  infinity  of  trans- 
formations— as  x-\-2y,  x-\-\y,  x  —  \y,  x+Vi^y,  and  so  on 
— all  of  them  similar  in  type,  and  a  corresponding  infinity 
of  transforms  (V)  or  (2')  of  the  same  expression  (1)  or 
equation  (2) ;  owing  to  the  invariance  of  b2  —  ac  under  each 
and  all  of  these  transformations,  we  know  that,  if  the 
roots  of  (2)  are  equal,  or  real  and  unequal,  or  imaginary, 
then  the  same  is  true  of  the  roots  of  each  one  of  the 
infinitely  many  transform  equations  (2').  I  have  ex- 
plained this  simple  fact, — small  indeed  as  a  mustard  seed, 
— so  fully  because,  as  already  said,  Lagrange's  observa- 
tion of  it  was  the  germ  of  the  now  vast  and  still  growing 
mathematical  theory  of  invariance.  Its  early  history 
owns  great  names:  Gauss  who  in  1801  showed  the  dis- 
criminant of  the  ternary  quadratic,  ax2-\-by2+cz2-{- 
2dxy  -\-2exz-\-2fyz,  to  be  invariant  under  the  transforma- 
tion replacing  x  by  Jx+By+Cz,  y  by  Dx+Ey  -\-Fz  and 
z  by  Gx-\-Ily  +/z;  Boole  who  in  1841  established,  among 
other  interesting  results,  the  invariance  of  the  discrimi- 
nant of  expressions  involving  an  arbitrary  number  of 
variables;  Cayley  who,  incited  by  Boole's  beautiful 
results,  assailed  the  problem  of  ascertaining  all  invariant 
functions  of  the  coefficients  of  an  equation  of  degree  11 
and  produced  in  rapid  succession  his  great  memoirs  on 


194  MATHEMATICAL   PHILOSOPHY 

Quantics;  Sylvester  who  soon  joined  Cayley  in  the  field, 
brilliantly  rivalled  his  researches  there  and  conceived  the 
subject  more  poetically  as  the  Calculus  of  Forms;  Eisen- 
stein,  Aronhold,  Hermite,  Clebsch,  Gordan  and  others; 
names  representing,  you  see,  both  Great  Britain  and  the 
continent  of  Europe.  The  result  is  a  truly  colossal 
doctrine  variously  styled  the  algebra  of  quantics,  the 
theory  of  algebraic  invariants  and  the  theory  or  calculus 
of  forms — a  doctrine  which  has  not  yet  ceased  to  grow  and 
to  which  American  mathematicians  have  recently  made 
valuable  contributions.  But  this  algebraic  theory  is  by 
no  means  the  whole  of  the  mathematical  doctrine  of 
invariance;  it  is  only  the  oldest  and  most  elaborate  part 
of  it;  every  division  of  mathematics  has  its  problem  of 
invariants;  and  vital  portions  of  many  subjects, — number 
theory,  for  example,  differential  equations,  various  func- 
tion theories,  all  the  multifarious  branches  of  geometry, — 
belong  to  the  doctrine  in  question. 

It  would  be  a  great  mistake  to  imagine  that  the  interest 
of  mathematicians  in  the  matter  of  invariance  is  peculiar 
to  them;  their  method  of  handling  it, — in  the  abstract 
by  logical  means, — is  indeed  peculiar  to  them;  but  the 
matter  itself  is  not;  for  a  little  reflection  suffices  to  show 
that  search  for  Constance, — constant  quantities,  constant 
properties,  constant  relations, — in  our  world  of  ceaseless 
change,  is  just  as  much  the  concern  of  religion,  of  philos- 
ophy, of  political  science,  of  education,  of  art,  and  of 
natural  science  as  it  is  of  mathematics,  though  each  of  the 
former  enterprises  has,  like  mathematics,  a  method  of  its 
own,  and  differs  from  its  allies  in  respect  to  the  type  of 
invariants  it  seeks. 

Consider  natural  science,  for  example.  What  is  it? 
For  our  present  purpose,  it  is  sufficiently  characterized  by 


INVARIANCE  195 

its  conscious  aim,  and  that  aim  is  discovery  of  those 
uniformities  in  the  course  of  Nature  which  men  of  natural 
science  are  wont  to  call  natural  laws.  What,  pray,  is  a 
natural  law?  A  natural  law, — if,  strictly  speaking,  there 
be  such  a  thing  outside  the  conception  thereof, — is 
fundamentally  nothing  more  nor  less  than  a  constant 
connection  among  inconstant  phenomena:  it  is,  in  other 
words,  an  invariant  relation  among  variant  terms.  It  is 
necessary  to  notice  the  sense  in  which  the  term  "  rela- 
tion "  is  here  used.  In  the  preceding  lecture,  where  we 
spoke  of  dyadic  relations,  it  was  said  that  such  a  relation, 
determined  by  a  propositional  function  F(x,  y)y  consists 
of  all  the  couples  (x,  y)  of  verifiers  of  the  function.  In 
accordance  with  that  view, — which  is  the  extensional  view 
of  relations, — the  relation  determined  by  the  function, 
x  is  the  father  of  y,  is  the  class  of  all  the  couples  (x,  y) 
such  that  the  x  is  some  male  that  begot  the  y,  and  all  such 
couples  are  regarded  as  coexisting  and  thus  constituting 
the  relation  once  for  all  even  though  most  fathers  and 
children  are  either  dead  or  unborn.  There  is,  however, 
another  view  of  relations, — the  intensional  view, — in  which 
the  concept  of  father  is  the  concept  of  a  constant  relation 
which  does  indeed  subsist  between  the  terms  of  such 
a  couple  if  and  when  the  latter  exists  but  which  would 
continue  to  exist  in  its  full  integrity  as  a  relation  at  an 
instant  or  during  an  interval  when  there  were  neither 
fathers  nor  children.  This  view  makes  it  possible  to 
speak,  in  a  "  natural  "  way,  of  a  relation  as  being  itself 
constant  while  having  in  the  flux  of  the  world  a  temporal 
succession  of  terms  or  sets  of  terms.  And  this  intensional 
or  intrinsic  sense  of  the  term  relation  is  the  sense  in 
which  I  employ  it  when  I  say  that  a  law  of  nature  is  simply 
an  invariant  relation  among  variant  terms. 


196  MATHEMATICAL   PHILOSOPHY 

In  this  conception  of  natural  law  and  the  consequent 
conception  of  natural  science  as  having  for  its  aim  discov- 
ery of  invariant  relations  among  the  things  that  appear  and 
disappear  in  the  flowing  pageant  of  the  world,  there  is,  I 
believe,  nothing  new  except  its  setting  and  its  manner. 
Rankine,  for  example,  in  a  paper  presented  before  the  Glas- 
gow Philosophical  Society  in  1867,  said:  "One  of  the  chief 
objects  of  mathematical  Physics  is  to  ascertain,  by  the 
help  of  experiment  and  observation,  what  physical  quan- 
tities are  conserved"  And  among  invariants  thus  found 
he  instances  mass,  resultant  momentum,  total  energy,  and 
other  things.  More  embracing  are  the  words  of  Major 
MacMahon  in  his  address  to  the  Mathematical  Section 
of  the  British  Association  in  1901.  "  In  any  subject  of 
inquiry  there  are,"  he  says,  "  certain  entities,  the  mutual 
relations  of  which,  under  various  conditions,  it  is  desirable 
to  ascertain.  A  certain  combination  of  these  may  be 
found  to  have  an  unalterable  value  where  the  entities  are 
submitted  to  certain  processes  or  are  made  the  subject 
of  certain  operations.  The  theory  of  invariants  in  its 
evident  scientific  meaning  determines  these  combinations, 
elucidates  their  properties,  and  expresses  results  when 
possible  in  terms  of  them.  Many  of  the  general  principles 
of  political  science  can  be  expressed  by  means  of  invarian- 
tive  relations  connecting  the  factors  which  enter  as  enti- 
ties into  the  special  problems.  The  great  principle  of 
chemical  science  which  asserts  that  when  elementary  or 
compound  bodies  combine  with  one  another,  the  total 
weight  of  the  material  is  unchanged,  is  another  case  in 
point.  Again,  in  Physics,  a  given  mass  of  gas  under  the 
operation  of  varying  pressure  and  temperature  has  the 
well-known  invariant,  pressure  multiplied  by  volume  and 
divided  by  absolute  temperature."     You  doubtless  know 


INVARIANCE  197 

that  similar  examples  might  be  cited  at  great  length; 
and  I  need  hardly  say  that,  if  some  of  those  cited  by 
Rankine  and  MacMahon  may  have  to  be  withdrawn  in 
view  of  recent  physical  refinements,  the  weight  and 
justice  of  their  main  contentions  remain  unimpaired. 

Interest  in  things  that  abide, — interest  in  stable 
values,  transcending  time  and  change, — is  as  fundamental 
and  regnant  in  art  as  in  natural  science.  It  is  true  that 
the  invariants  which  art  seeks  in  its  own  way  to  find  and 
in  its  own  way  to  disclose  or  represent  are  not  sharply 
defined;  like  personal  identity,  they  are  of  the  class  of 
those  which  a  little  while  ago  we  described  as  unformu- 
lated or  qualitative  invariants;  they  are  none  the  less 
genuine  invariants,  and  no  defect  of  their  definition  can 
disguise  or  dim  the  fact  that,  like  natural  science  and  like 
mathematics,  art, — art  in  its  great  moods  and  proper 
character  as  art, — contemplates  the  world  under  the 
aspect  of  eternity,  aims  at  what  is  permanent  in  the 
"  fleeting  show,"  devotes  itself  to  goods  that  are  ever- 
lasting. For  the  fact  is  manifest  in  many  ways.  A 
thing  of  beauty  is  a  joy  forever.  Who  does  not  know,  or 
at  all  events  feel,  the  deep  and  proper  meaning  of  this 
familiar  mot?  It  is  not  that  any  phrase  or  picture  or  poem 
or  symphony  or  statue  or  temple  will  escape  the  doom  of 
temporal  things;  nor  that  the  joy  they  may  give  you  or 
me  will  endure;  it  is  that  a  certain  quality, — the  quality 
in  virtue  of  which  a  thing  of  beauty  is  such  a  thing, — is 
timeless,  unbegotten  and,  though  its  temporal  embodi- 
ments must  perish,  is  itself  imperishable.  "  The  purpose 
of  art,"  said  John  LaFarge,  "  is  commemoration."  In 
ceternitatum  pingo,  said  Zeuxis,  the  Greek  painter.  One 
need  not  be  an  artist  to  understand  that,  in  the  words  of 
Joshua  Reynolds,  "  The  idea  of  beauty  in  each  species  of 


198  MATHEMATICAL   PHILOSOPHY 

being  is  perfect,  invariable,  divine."  One  need  not  be  a 
Plato,  to  know,  as  I  have  elsewhere  said,  "  that  by  a 
faculty  of  imaginative,  mystical,  idealizing  discernment 
there  is  revealed  to  us,  amid  the  fleeting  beauties  of  Time, 
the  immobile  presence  of  Eternal  beauties,  immutable 
archetypes  and  source  of  the  grace  and  loveliness  beheld 
in  the  shifting  scenes  of  the  flowing  world  of  sense." 
These  archetypes, — perfect,  unbegotten,  everlasting, — 
these  are  the  invariants  which  it  is  the  aim  and  the 
function  of  art  to  discern  and  to  represent. 

It  seems  unnecessary  to  argue  here  that  what  has  been 
said  respecting  the  motivity  of  natural  science  and  the 
like  motivity  of  art  is,  mutatis  mutandis,  equally  valid  in 
education,  in  jurisprudence,  in  political  science,  in  eco- 
nomics, in  philosophy,  and  in  religion,  for  the  sufficient 
evidence  is  not  far  to  seek  and  you  have,  I  trust,  an  ample 
clue. 

And  so  we  are  led  to  a  grave  and  impressive  thesis — 
a  thesis  regarding  the  principle  which  unites  all  the  great 
forms  of  human  endeavor.  The  thesis  is  that  the  unifying 
principle,— the  central  binding  thread  of  human  history, — 
the  tie  of  comradeship  among  the  spiritual  enterprises  of 
man, — is  passion  and  search  for  things  eternal:  the  thesis 
is  that  quest  of  invariance, — quest  of  abiding  reality, — is 
itself  the  sovereign  invariant  in  the  changeful  life  of  reason. 

You  are  students  of  philosophy — students  of  the  life  of 
reason.  To  you,  therefore,  with  the  utmost  confidence 
I  commend  the  thesis  as  worthy  of  your  best  attention. 
As  you  meditate  upon  it,  there  will  arise  within  you  the 
bright  vision  of  a  great  and  inspiring  task — a  task  that 
has  not  been  performed  nor  even  essayed.  I  mean  the 
writing  of  A  Critical  History  of  Thought  Viewed  as  the 
Quest  of  Invariance  in  a  Fluctuant  World.  Taking 
Thought  in  its  widest  sense,  embracing  all  the  cardinal 


INVARIANCE  199 

enterprises  of  the  human  spirit,  such  a  history,  if  ever  it 
be  written,  will  have  a  scope  greatly  exceeding  that  of  any 
extant  "  history  of  philosophy  ";  in  addition  to  that  and 
far  more  important,  it  will  have,  unlike  such  histories 
of  philosophy,  a  natural  unity,  for  it  will  have  a  unity 
derived  from  the  unity  of  Thought  itself.  The  history  of 
Thought  in  our  ever-growing,  ever-perishing  universe  is 
the  history  of  human  endeavor  to  answer  the  question: 
What  abides?  The  task  of  criticism  as  thus  conceived 
is  indeed  immense.  What  abides?  To  cbllate  and  name 
and  locate  and  order;  to  understand,  describe  and  explain; 
to  compare,  judge  and  evaluate  all  of  the  chief  responses 
that  the  religions  and  arts  and  sciences  and  philosophies 
and  speculations  of  the  ages  have  made  to  the  question 
in  divers  tongues — these  are  the  things  which  constitute 
and  define  the  obligations  of  the  historian  of  Thought. 
If  you  be  ambitious — but  what  is  ambition?  Some 
one  has  conceived  it  to  be  a  great  man's  desire  to  cast  his 
shadow  endlessly  down  the  course  of  history.  I  prefer 
to  regard  it  as  the  urge  felt  by  great  men  to  exercise  the 
power  of  creation;  for  this  power,  the  power  of  creative 
love — peculiar  to  man — is  the  power  which,  inheriting 
civilization  as  fruit  of  dead  men's  toil,  receives  it,  not 
as  the  beasts  receive  the  natural  fruits  of  earth,  but  as 
spiritual  capital  to  be  more  and  more  augmented — with 
ever-increasing  speed  in  the  course  of  successive  genera- 
tions— for  the  well-being  of  humankind  including  pos- 
terity. If  you  be  ambitious  in  that  sense,  the  task  I  have 
tried  to  signalize  is  worthy  of  your  mettle — worthy  of 
whatever  genius  you  may  have,  of  all  the  learning  you  can 
acquire,  of  all  your  talent  for  devotion  and  toil.  Let  me 
say  finally,  as  I  have  already  intimated,  that  the  bearings 
thereupon  of  the  mathematical  theory  of  invariance  are 
the  bearings  of  a  prototype  and  guide. 


LECTURE  XII 
The  Group  Concept 

THE     NOTION    SIMPLY    EXEMPLIFIED    IN    MANY    FIELDS IS 

"MIND"    A    GROUP? GROUPS    AS    INSTRUMENTS    FOR 

DELIMITING  DOCTRINES — CONNECTION  OF  GROUP  WITH 
TRANSFORMATION  AND  INVARIANCE THE  IDEA  FORE- 
SHADOWED     IN      THE      AGES      OF      SPECULATION THE 

PHILOSOPHY     OF     THE     COSMIC     YEAR THE     IDEA     OF 

PROGRESS. 

You  will  recall  that  near  the  close  of  the  introductory 
lecture  I  gave  a  partial  list  of  those  mathematical  terms 
which  may  be  rightly  regarded  as  denoting  the  pillar- 
concepts  of  the  science.  Among  these  are  function,  rela- 
tion, transformation,  invariance  and  group.  In  Lecture  X 
we  saw  that  the  first  three  denote  three  aspects  of  one  and 
the  same  thing  seen  from  different  points  of  view,  and 
this  thing, — whether  we  call  it  function  or  relation  or 
transformation, — is  sovereign — the  central  support  not 
only  of  mathematics  but  of  the  entire  edifice  of  science 
taken  in  the  widest  sense.  In  the  closest  logical  connec- 
tion therewith  stand  the  two  great  concepts  of  invariance 
and  group;  so  that  I  can  hardly  overemphasize  the  impor- 
tance of  your  learning  to  associate  the  three  notions  as 
intimately  in  your  thought  as  they  are  associated  in  fact: 
Transformation  —  Invariance  —  Group.  Of  transforma- 
tion I  endeavored  to  give  some  account  in  Lecture  X  and 

200 


THE   GROUP  CONCEPT  201 

of  invariance  in  Lecture  XI.  I  invite  your  attention 
during  the  present  hour  to  the  notion  of  group.  Even 
if  I  were  a  specialist  in  group  theory, — which  I  am  not, — 
I  could  not  in  one  hour  give  you  anything  like  an  extensive 
knowledge  of  it,  nor  facility  in  its  technique,  nor  a  sense 
of  its  intricacy  and  proportions  as  known  to  its  devotees, 
the  priests  of  the  temple.  But  the  hour  should  suffice  to 
start  you  on  the  way  to  acquiring  at  least  a  minimum  of 
what  a  respectable  philosopher  should  know  of  this  funda- 
mental subject;  and  such  a  minimum  will  include:  a 
clear  conception  of  what  the  term  "group"  means; 
ability  to  illustrate  it  copiously  by  means  of  easily  under- 
stood examples  to  be  found  in  all  the  cardinal  fields  of 
interest — number,  space,  time,  motion,  relation,  play, 
work,  the  world  of  sense-data  and  the  world  of  ideas; 
a  glimpse  of  its  intimate  connections  with  the  ideas  of 
transformation  and  invariance;  an  inkling  of  it  both  as 
subject-matter  and  as  an  instrument  for  the  delimitation 
and  discrimination  of  doctrines;  and  discernment  of  the 
concept  as  vaguely  prefigured  in  philosophic  speculation 
from  remote  antiquity  down  to  the  present  time. 

I  believe  that  the  best  way  to  secure  a  firm  hold  of  the 
notion  of  group  is  to  seize  upon  it  first  in  the  abstract 
and  then,  by  comparing  it  with  concrete  examples, 
gradually  to  win  the  sense  of  holding  in  your  grasp  a 
living  thing.  In  presenting  the  notion  of  group  in  the 
abstract,  it  is  convenient  to  use  the  term  system.  This 
term  has  many  meanings  in  mathematics  and  so  at  the 
outset  we  must  clearly  understand  the  sense  in  which 
the  term  is  to  be  employed  here.  The  sense  is  this:  as 
employed  in  the  definition  of  group,  the  term  system 
means  some  definite  class  of  things  together  with  some 
definite    rule,    or   way,    in    accordance   with   which    any 


202  MATHEMATICAL   PHILOSOPHY 

member  of  the  class  can  be  combined  with  any  member 
of  it  (either  with  itself  or  any  other  member).  For  a 
simple  example  of  such  a  system  we  may  take  for  the  class 
the  class  of  ordinary  whole  numbers  and  for  the  rule  of  com- 
bination the  familiar  rule  of  addition.  You  should  note 
that  there  are  three  and  only  three  respects  in  which  two 
systems  can  differ :  by  having  different  classes,  by  having 
different  rules  of  combination,  and  by  differing  in  both 
of  these  ways. 

The  definition  of  the  term  "  group  "  is  as  follows. 

Let  S  denote  a  system  consisting  of  a  class  C  (whose 
members  we  will  denote  by  a>  b,  c  and  so  on)  and  of  a  rule 
of  combination  (which  rule  we  will  denote  by  the  symbol  o, 
so  that  by  writing,  for  example,  aob,  we  shall  mean  the 
result  of  combining  b  with  a).  The  system  S  is  called  a 
group  if  and  only  if  it  satisfies  the  following  four  conditions : 

(a)  If  a  and  b  are  members  of  C,  then  aob  is  a  member 
of  C;  that  is,  aob  =c,  where  c  is  some  member  of  C. 

(b)  If  a,  by  c  are  members  of  C,  then  (aob)oc  =  ao(boc); 
that  is,  combining  c  with  the  result  of  combining  b  with  a 
yields  the  same  as  combining  with  a  the  result  of  combin- 
ing c  with  b;  that  is,  the  rule  of  combination  is  asso- 
ciative. 

(c)  The  class  C  contains  a  member  i  (called  the 
identical  member  or  element)  such  that  if  a  be  a  member 
of  C,  then  aoi  =  ioa=a;  that  is,  C  has  a  member  such 
that,  if  it  be  combined  with  any  given  member,  or  that 
member  with  it,  the  result  is  the  given  member. 

(d)  If  a  be  a  member  of  C,  then  there  is  a  member  a' 
(called  the  reciprocal  of  a)  such  that  aoa'  =a'oa  =i;  that 
is,  each  member  of  C  is  matched  by  a  member  such  that 
combining  the  two  gives  the  identical  member. 

Other   definitions   of  the  term   "  group  "   have   been 


THE  GROUP  CONCEPT  203 

proposed  and  are  sometimes  used.  The  definitions  are 
not  all  of  them  equivalent  but  they  all  agree  that  to  be  a 
group  a  system  must  satisfy  condition  (a). 

Systems  satisfying  condition  (a)  are  many  of  them  on 
that  account  so  important  that  in  the  older  literature  of 
the  subject  they  are  called  groups,  or  closed  systems,  and 
are  now  said  to  have  "  the  group  property,"  even  if  they 
do  not  satisfy  conditions  (&),  (c)  and  (d).  The  propriety 
of  the  term  "  closed  system  "  is  evident  in  the  fact  that  a 
system  satisfying  (a)  is  such  that  the  result  of  combining 
any  two  of  its  members  is  itself  a  member — a  thing  in 
the  system,  not  out  of  it. 

Various  Simple  Examples  of  Groups  and  of  Systems 
that  Are  Not  Groups. — You  observe  that  by  the  foregoing 
definition  of  group  every  group  is  a  system;  groups,  as 
we  shall  see,  are  infinitely  numerous;  yet  it  is  true  that 
relatively  few  systems  are  groups  or  have  even  the  group 
property — so  few  relatively  that,  if  you  select  a  system  at 
random,  it  is  highly  probable  you  will  thus  hit  upon  one 
that  is  neither  a  group  nor  has  the  group  property. 

Take,  for  example,  the  system  S\  whose  class  C  is  the 
class  of  integers  from  I  to  10  inclusive  and  whose  rule  of 
combination  is  that  of  ordinary  multiplication  X;  3  X4  =  12; 
12  is  not  a  member  of  C,  and  so  Si  is  not  closed — it  has 
not  the  group  property. 

Let  S2  have  for  its  C  the  class  of  all  the  ordinary 
integers,  1,  2,  3,  .  .  .  ad  infinitum,  and  let  o  be  X  as 
before;  as  the  product  of  any  two  integers  is  an  integer, 
(a)  is  satisfied — S2  is  closed,  has  the  group  property; 
(b),  too,  is  evidently  satisfied,  and  so  is  (c),  the  identity 
element  being  1  for,  if  n  be  any  integer,  wXi  =1  Xn  =n\ 
but  (d)  is  not  satisfied — none  of  the  integers  (except  1) 
composing  C  has  a  reciprocal  in  C — there  is,  for  example, 


204  MATHEMATICAL   PHILOSOPHY 

no  integer  n  such  that  2X«=«X2  =  i;  and  so  S2,  though 
it  has  the  group  property,  is  not  a  group. 

Let  S3  be  the  system  consisting  of  the  class  C  of  all  the 
positive  and  negative  integers  including  zero  and  of 
addition  as  the  rule  of  combination;  you  readily  see  that 
S3  is  a  group,  zero  being  the  identical  element,  and  each 
element  having  its  own  negative  for  reciprocal. 

A  group  is  said  to  be  finite  or  infinite  according  as  its 
C  is  a  finite  or  an  infinite  class  and  it  is  said  to  be  Abelian 
or  non-Abelian  according  as  its  rule  of  combination  is  or 
is  not  commutative — according,  that  is,  as  we  have  or  do 
not  have  aob=boa,  where  a  and  b  are  arbitrary  members 
of  C.  You  observe  that  the  group  S3  is  both  infinite  and 
Abelian. 

For  an  example  of  a  group  that  is  finite  and  Abelian 
it  is  sufficient  to  take  the  system  S4  whose  C  is  composed 
of  the  four  numbers,  1,  —I,  i,  —i,  where  i  is  V~^i,  and 
whose  rule  of  combination  is  multiplication;  you  notice 
that  the  identical  element  is  1,  that  1  and  —  1  are  each 
its  own  reciprocal  and  that  i  and  —i  are  each  the  other's 
reciprocal. 

Let  S5  have  the  same  C  as  S3  and  suppose  o  to  be  sub- 
traction instead  of  addition;  show  that  S5  has  the  group 
property  but  is  not  a  group.  Show  the  like  for  Sfi  in 
which  C  is  the  same  as  before  and  o  denotes  multiplica- 
tion. Show  that  S7  where  C  is  the  same  as  before  and  o 
means  the  rule  of  division,  has  not  even  the  group  property. 

Consider  Sg  where  C  is  the  class  of  all  the  rational 
numbers  (that  is,  all  the  integers  and  all  the  fractions 
whose  terms  are  integers,  it  being  understood  that  zero 
can  not  be  a  denominator)  and  where  o  denotes  +;  you 
will  readily  find  that  Ss  is  a  group,  infinite  and  Abelian. 
Examine  the  systems  obtained  by  keeping  the  same  C 


THE   GROUP   CONCEPT  205 

and  letting  o  denote  subtraction,  then  multiplication,  then 
division.     Devise  a  group  system  where  o  means  division. 

If  S  and  S'  be  two  groups  having  the  same  rule  of 
combination  and  if  the  class  C  of  S  be  a  proper  part  of  the 
class  C  of  S'  {i.e.,  if  the  members  of  C  are  members  of 
C  but  some  members  of  C  are  not  in  C),  then  S  is  said 
to  be  a  sub-group  of  S'.  Observe  that  S3  is  a  sub- 
group of  Sg. 

Show  that  S9  is  a  group  if  its  C  is  the  class  of  all  real 
numbers  and  its  o  is  +;  note  that  Sg  is  a  sub-group  of 
S9  and  hence  that  S3  is  a  sub-group  of  a  sub-group  of  a 
group.  Is  So  itself  a  sub-group?  If  so,  of  what  group  or 
groups?  Examine  the  systems  derived  from  S0  by 
altering  the  rule  of  combination. 

The  most  difficult  thing  that  teaching  has  to  do  is  to 
give  a  worthy  sense  of  the  meaning  and  scope  of  a  great 
idea.  A  great  idea  is  always  generic  and  abstract  but  it 
has  its  living  significance  in  the  particular  and  concrete — 
in  a  countless  multitude  of  differing  instances  or  examples 
of  it;  each  of  these  sheds  only  a  feeble  light  upon  the  idea, 
leaving  the  infinite  range  of  its  significance  in  the  dark; 
whence  the  necessity  of  examining  and  comparing  a  large 
number  of  widely  difFering  examples  in  the  hope  that  many 
little  lights  may  constitute  by  union  something  like  a 
worthy  illustration;  but  to  present  these  numerous 
examples  requires  an  amount  of  time  and  a  degree  of 
patience  that  are  seldom  at  one's  disposal,  and  so  it  is 
necessary  to  be  content  with  a  selected  few.  And  now 
here  is  the  difficulty:  if  the  examples  selected  be  complex 
and  difficult,  they  repel;  if  they  be  simple  and  easy,  they 
are  not  impressive;  in  either  case,  the  significance  of  the 
general  concept  in  question  remains  ungrasped  and  unap- 
preciated.    I  am  going,  however,  to  take  the  risk — to  the 


206  MATHEMATICAL    PHILOSOPHY 

foregoing  illustrations  of  the  group  concept  I  am  going 
to  add  a  few  further  ones, — some  of  them  very  simple, 
some  of  them  more  complex, — trusting  that  the  former 
may  not  seem  to  you  too  trivial  nor  the  latter  too  hard. 

Every  one  has  seen  the  pretty  phenomenon  of  a  grey 
squirrel  rapidly  rotating  a  cylindrical  wire  cage  enclosing 
it.  It  may  rotate  the  cage  in  either  of  two  opposite  ways, 
senses  or  directions.  Let  us  think  of  rotation  in  only  one 
of  the  ways,  and  let  us  call  any  rotation,  whether  it  be 
much  or  little,  a  turn.  Each  turn  carries  a  point  of  the 
cage  along  a  circle-arc  of  some  length,  short  or  long. 
Denote  by  R  the  special  turn  (through  3600)  that  brings 
each  point  of  the  cage  back  to  its  starting  place.  Let 
Sio  be  the  system  whose  C  is  the  class  of  all  possible 
turns  and  whose  o  is  addition  of  turns  so  that  aob  shall 
be  the  whole  turn  got  by  following  turn  a  by  turn  b. 
You  see  at  once  that  S  has  the  group  property  for  the  sum 
of  any  two  turns  is  a  turn;  it  is  equally  evident  that  the 
associative  law — condition  (b) — is  satisfied.  Note  that  R 
is  equivalent  to  no  turn, — equivalent  to  rest, — equivalent 
to  a  zero  turn,  if  you  please;  note  that,  if  a  be  a  turn 
greater  than  R  and  less  than  2R,  then  a  is  equivalent  to 
as  excess  over  R;  that,  if  a  be  greater  than  zR  and  less 
than  3R,  then  a  is  equivalent  to  as  excess  over  2R;  and 
so  on;  thus  any  turn  greater  than  R  and  not  equal  to  a 
multiple  of  R  is  equivalent  to  a  turn  less  than  R;  let  us 
regard  any  turn  that  is  thus  greater  than  R  as  identical 
with  its  equivalent  less  than  R;  we  have,  then,  to  con- 
sider no  turns  except  R  and  those  less  than  R — of  which 
there  are  infinitely  many;  you  see  immediately  that,  if  0 
be  any  turn,  aoR=Roa—a,  which  means  that  condition 
(c)  is  satisfied  with  R  for  identical  element.  Next  notice 
that  for  any  turn  a  there  is  a  turn  a'  such  that  ooa!  — 


THE   GROUP  CONCEPT  207 

a'oa=R.  Hence  Sio  is,  as  you  see,  a  group.  Show  it 
to  be  Abelian.  You  will  find  it  instructive  to  examine 
the  system  derived  from  Sio  by  letting  C  be  the  class  of 
all  turns  (forward  or  backward). 

Perhaps,  you  will  consider  the  system  suggested  by  the 
familiar  spectacle  of  a  ladybug  or  a  measuring-worm 
moving  round  the  rim  or  edge  of  a  circular  tub;  or  the 
system  suggested  by  motions  along  the  thread  of  an 
endless  screw;  or  that  suggested  by  the  turns  of  the  earth 
upon  its  axis;  or  that  suggested  by  the  motions  of  a  planet 
in  its  orbit. 

Do  such  examples  give  the  meaning  of  the  group 
concept?  Each  one  gives  it  somewhat  as  a  water-drop 
gives  the  meaning  of  ocean,  or  a  burning  match  the  mean- 
ing of  the  sun,  or  a  pebble  the  meaning  of  the  Rocky 
Mountains.  Are  they,  therefore,  to  be  despised?  Far 
from  it.  Taken  singly,  they  tell  you  little;  but  taken 
together,  if  you  allow  your  imagination  to  play  upon 
them,  noting  their  differences,  their  similitudes,  and  the 
variety  of  fields  they  represent,  they  tell  you  much. 
Let  us  pursue  them  further,  having  a  look  in  other  fields. 

Consider  the  field  of  the  data  of  sense, — a  field  of  uni- 
versal interest, — and  fundamental.  We  are  here  in  the 
domain  of  sights  and  sounds  and  motions  among  other 
things.  Are  there  any  groups  to  be  found  here?  Who, 
except  the  blind-born,  are  not  lovers  of  color?  Do  the 
colors  constitute  a  group?  I  mean  sensations  of  color, — 
color  sensations, — including  all  shades  thereof  and  white 
and  black.  Denote  by  Su  the  system  whose  C  is  the  class 
of  all  such  sensations  and  whose  rule  of  combination  is, 
let  us  say,  the  mixing  of  such  sensations.  But  what  are 
we  to  understand  by  the  mixing  of  two  color  sensations? 
Suppose  we  have  two  small   boxes  of  powder, — say   of 


208  MATHEMATICAL   PHILOSOPHY 

finely  pulverized  chalk, — a  box  of,  say,  red  powder  and 
a  box  of  blue;  one  of  the  powders  gives  us  the  sensation 
red,  the  other  the  sensation  blue;  let  us  thoroughly  mix 
the  powders;  the  mixture  gives  us  a  color  sensation; 
we  agree  to  say  that  we  have  mixed  the  sensations  and 
that  the  new  sensation  is  the  result  of  mixing  the  old 
ones.  As  the  combination  of  any  two  color  sensations  is 
a  color  sensation,  Sn  has,  you  see,  the  group  property. 
Is  it  a  group?  Evidently  condition  (b)  is  satisfied.  Are 
conditions  (c)  and  (d)  also  satisfied  ? 

Let  us  pass  from  colors  to  figures  or  shapes, — to  figures 
or  shapes,  I  mean,  of  physical  or  material  objects, — rocks, 
chairs,  trees,  animals  and  the  like, — as  known  to  sense- 
perception.  No  doubt  what  we  ordinarily  call  perception 
of  an  object's  figure  or  shape  is  genetically  complex,  a 
result  of  experience  contributed  to  by  two  or  more  senses, 
as  sight,  touch,  motion;  let  us  not,  however,  try  to 
analyze  it  thus;  let  us  take  it  at  its  face  value — let  us 
regard  it  as  being,  what  it  appears  to  be  before  analytic 
reflection  upon  it,  a  sense-given  datum;  and  let  us  confine 
ourselves  to  the  sense  of  sight.  Here  is  a  dog;  its  ears 
have  shape;  so,  too,  its  eyes,  its  nose  and  the  other 
features  of  its  head;  these  shapes  combine  to  make  the 
shape  or  figure  of  the  head;  each  other  one  of  its  visible 
organs  has  a  shape  of  its  own;  these  shapes  all  of  them 
combine  to  make  that  thing  which  we  call  the  shape  or 
figure  of  the  dog.  Yonder  is  a  table;  it  has  a  shape,  and 
this  is  due  to  some  sort  of  combination  of  the  shapes  of  its 
parts — legs,  top,  and  so  on;  upon  it  are  several  objects — 
a  picture  frame,  a  candlestick,  some  vases;  each  has  a 
shape;  the  table  and  the  other  things  together  constitute 
one  object — disclosed  as  such  to  a  single  glance  of  the  eye; 
this  object  has  a  figure  or  shape  due  to  the  combined  pres- 


THE  GROUP  CONCEIT  209 

ence  of  the  other  shapes.  In  speaking  of  the  dog  and  the 
table,  I  have  been  using  the  word  "  combination  "  in  a 
very  general  sense.  Can  it,  in  this  connection,  be  made 
precise  enough  for  our  use?  Is  it  possible  to  find  or  frame 
a  rule  by  which,  any  two  visible  shapes  being  given,  these 
can  be  combined?  If  so,  is  the  result  of  the  combination 
a  visible  shape?  If  so,  the  system  consisting  of  the  rule 
and  the  total  class  of  shapes  has  the  group  property. 
Does  the  system  satisfy  the  remaining  three  conditions 
for  a  group  ? 

And  what  of  sounds — sensations  of  sound  ?  Are  sounds 
combinable?  Is  the  result  always  a  sound  or  is  it  some- 
times silence?  If  we  agree  to  regard  silence  as  a  species 
of  sound, — as  the  zero  of  sound, — has  the  system  of  sounds 
the  property  of  a  group?  There  is  the  question  of  thresh- 
olds: sound  is  a  vibrational  phenomenon;  if  the  rate  of 
vibration  be  too  slow  or  too  great, — say,  100,000  per 
second, — no  sound  is  heard.  If  you  disregard  the  thresh- 
olds, has  the  sound  system  the  group  property?  Is  it  a 
group?  If  so,  what  is  the  identical  element?  And  what 
would  you  say  is  the  reciprocal  of  a  given  sound  or  tone? 

Consider  other  vibrational  phenomena — as  those  of 
light  or  electricity.  Can  you  so  conceive  them  as  to  get 
group  systems?  Sharpen  your  questions  and  then  carry 
them  to  physicists.  You  need  have  no  hesitance — the 
service  is  apt  to  be  mutual. 

The  Infinite  Abelian  Group  of  Angel  Flights. — We  are 
accustomed  to  think  of  ourselves  as  being  in  a  boundless 
universe  of  space  filled  with  what  we  call  points  any  two 
of  which  are  joined  by  what  we  call  a  straight  line. 
Imagine  one  of  those  curious  creatures  which  are  to-day 
for  most  of  us  hardly  more  than  figures  of  speech  but 
which  for  many  hundreds  of  years  were  very  real  and  very 


210  MATHEMATICAL   PHILOSOPHY 

lovely  or  very  terrible  things  for  millions  of  men,  women 
and  children  and  were  studied  and  discoursed  about 
seriously  by  men  of  genius :  I  mean  angels.  Angels  can 
fly.  Let  us  confine  their  flights  to  straight  lines  but 
impose  no  other  restrictions.  I  am  going  to  ask  you  to 
understand  a  flight  as  having  nothing  but  length,  direction 
and  sense;  if  it  is  parallel  to  a  given  straight  line,  it  has 
that  line's  direction;  if  it  goes  from  A  towards  B,  it  has 
that  sense;  if  from  B  toward  A,  the  opposite  sense;  a 
flight  from  A  to  B  and  one  from  C  to  D  are  the  same  if 
they  agree  in  length,  direction  and  sense.  Consider  a 
flight  a  from  point  A  to  point  B  followed  by  a  flight  b 
from  B  to  C;  you  readily  see  that  a  and  b  are  two  adjacent 
sides  of  a  parallelogram,  one  of  whose  diagonals  is  the 
direct  flight  d  from  A  to  C;  d  is  called  the  resultant  or 
flight-sum  of  a  and  b  because  d  tells  us  how  far  the  angel 
has  finally  got  from  the  starting  place;  and  so  we  write 
aob=d.  If  flight  V  goes  from  P  to  Q  but  agrees  with  b 
in  length,  direction  and  sense,  we  write  aob'  =d  as  before 
for,  as  already  said,  b  and  b'  are  one  and  the  same.  Now 
let  S12  denote  the  system  whose  C  is  the  class  of  all 
possible  angel  flights  including  rest,  or  zero  flight,  and 
whose  rule  of  combination  is  flight  summation  as  above 
explained;  you  see  at  once  that  Si2is  a  closed  system,  has 
the  group  property,  for  the  combination  of  any  two 
flights  is  a  flight;  if  a,  b  and  c  be  three  flights,  we  may  sup- 
pose them  to  go  respectively  from  A  to  B,  from  B  to  C, 
and  from  C  to  D;  consider  (aob)oc;  aob=d,  the  flight 
from  A  to  C;  doc=e,  the  flight  from  A  to  D;  now  con- 
sider ao(boc);  boc=d',  flight  from  B  to  D;  aod'=ef, 
flight  from  A  to  D;  so  e=e'  and  (aob)oc  =  ao(boc) ;  hence 
summation  of  flights  is  associative — condition  (b)  is  satis- 
fied.    Condition  (c)  is  satisfied  with  zero  (0)  flight  for  i; 


THE  GROUP  CONCEPT  211 

for,  if  a  be  any  flight,  it  is  plain  that  <zo0=0o#=tf.  And 
condition  id)  is  satisfied  for  it  is  evident  that  aoa'  = 
afoa=0  where  a'  and  a  agree  in  length  and  direction  but 
are  opposite  in  sense.  Hence  the  system  of  angel  flights 
is  a  group.  And  it  is  easy  to  see  that  it  is  both  infinite 
and  Abelian. 

What  I  have  here  called  an  angel  flight  is  known  in 
mathematics  and  in  physics  as  a  vector;  a  vector  has  no 
position — it  has  its  essential  and  complete  being  in  having 
a  length,  a  direction  and  a  sense.  And  so,  you  see,  the 
system  composed  of  the  vectors  of  space  and  of  vector 
addition  as  a  rule  of  combination  is  an  infinite  Abelian 
group. 

Connection  of  Groups  with  Transformations  and  Invari- 
ants.— Let  us  have  another  look  at  our  angel  flights,  or 
vectors.  I  am  going  to  ask  you  to  view  them  in  another 
light.  Let  V  be  any  given  vector — that  is,  a  vector  of 
given  length,  sense  and  direction;  where  does  it  begin 
and  where  does  it  end?  A  moment's  reflection  will  show 
you  that  every  point  in  the  universe  of  space  is  the  begin- 
ning of  a  vector  identical  with  V  and  the  end  of  a  vector 
identical  with  V.  Though  these  vectors  are  but  one,  it  is 
convenient  to  speak  of  them  as  many  equal  vectors — 
having  the  same  length,  direction  and  sense.  Let  the 
point  P  be  the  beginning  of  a  V  and  let  the  point  Q  be 
its  end.  Let  us  now  associate  every  such  P  with  its 
Q(P  — >  Q);  we  have  thus  transformed  our  space  of  points 
into  itself  in  such  wise  that  the  end  of  each  V  is  the 
transform  of  its  beginning;  call  the  transformation  7'; 
let  us  follow  it  with  a  transformation  T'  converting  the 
beginnings  of  all  vectors  equal  to  a  given  vector  V  into 
their  corresponding  ends.  What  is  the  result?  Notice 
that  T  converted  P  into  Q  and  that  T'  then  converted  Q 


212  MATHEMATICAL   PHILOSOPHY 

into  Q\  the  end  of  the  V  beginning  at  Q;  now  there  is  a 
vector  beginning  at  P  and  running  direct  to  Q'\  and  so 
there  is  a  transformation  T"  converting  P  into  Q' \  it  is 
this  T"  that  we  shall  mean  by  7b V .  Without  further 
talk,  you  see  that  our  group  of  angel  flights,  or  vectors, 
now  appears  as  an  infinite  Abelian  group  of  transformations 
(of  our  space  of  points  into  itself).  Such  transformations 
do  not  involve  motion  in  fact;  it  is  customary,  however, 
for  mathematicians  to  call  them  motions,  or  translations, 
of  space;  5',  for  example,  being  thought  of  and  spoken 
of  as  a  translation  of  the  whole  of  space  (as  a  rigid  body) 
in  the  direction  and  sense  of  V  and  through  a  distance 
equal  to  Vs  length.  In  accordance  with  this  stimulating, 
albeit  purely  figurative,  way  of  speaking,  the  group  in 
question  is  the  group  of  the  translations  of  our  space. 

We  are  now  in  a  good  position  to  glimpse  the  very 
intimate  connection  between  the  idea  of  group  and  the 
idea  of  invariance.  Suppose  we  are  given  a  group  of 
transformations;  one  of  the  big  questions  to  be  asked 
regarding  it  is  this:  what  things  remain  unaltered, — 
remain  invariant, — under  each  and  all  the  transformations 
of  the  group  ?  In  other  words,  what  property  or  proper- 
ties are  common  to  the  objects  transformed  and  their 
transforms  ?  Well,  we  have  now  before  us  a  certain  group 
of  space  transformations — the  group  of  translations. 
Denote  it  by  G.  Each  translation  in  G  converts  (trans- 
forms, carries,  moves)  any  point  into  a  point,  and  so 
converts  any  configuration  F  of  points,— any  geometric 
figure, — into  some  configuration  F'.  What  remains  un- 
changed? What  are  the  invariants?  It  is  obvious  that 
one  of  the  invariants, — a  very  important  one, — is  distance; 
that  is,  if  P  and  Q  be  any  two  points  and  if  their  trans- 
forms under  some  translation  be  respectively  P'  and  Q'> 


THE  GROUP  CONCEPT  213 

then  the  distance  between  P'  and  Q'  is  the  same  as  that 
between  P  and  Q;  another  is  order  among  points— if  Q 
is  between  P  and  R,  Q'  is  between  P'  and  R';  you  see 
at  once  that  angles,  areas,  volumes,  shapes  are  all  of  them 
unchanged:  in  a  word,  congruence  is  invariant- — if,  a 
translation  convert  a  figure  F  into  a  figure  F',  F  and  F' 
are  congruent.  Of  course  congruence  is  invariant  under 
all  the  translations  having  a  given  direction.  Do  these 
constitute  a  group?  Obviously  they  do,  and  this  group 
G'  is  a  sub-group  of  G.  Congruence  is  invariant  under 
G'\  it  is  also  invariant  under  G;  G'  is  included  in  G;  it  is 
natural,  then,  to  ask  whether  G  itself  may  not  be  included 
in  a  still  larger  group  having  congruence  for  an  invariant. 
The  question  suggests  the  inverse  of  the  one  with  which 
we  set  out.  The  former  question  was:  given  a  group, 
what  are  its  invariants?  The  inverse  question  is:  given 
an  invariant,  what  are  its  groups  and  especially  its  largest 
group?  This  question  is  as  big  as  the  other  one.  Con- 
sider the  example  in  hand.  The  given  invariant  is  con- 
gruence. Is  G, — the  group  of  translations, — the  largest 
group  of  space  transformations  leaving  congruence  un- 
changed? Evidently  not;  for  think  of  those  space  trans- 
formations that  consist  in  rotations  of  space  (as  a  rigid 
body)  about  a  fixed  line  (as  axis);  if  such  a  rotation 
converts  a  figure  F  into  F',  the  two  figures  are  congruent. 
It  is  clear  that  the  same  is  true  if  a  transformation  be  a 
twist — that  is,  a  simultaneous  rotation  about,  and  transla- 
tion along,  a  fixed  line.  All  such  rotations  and  twists 
together  with  the  translations  constitute  a  group  called 
the  group  of  displacements  of  space;  it  includes  all  trans- 
formations leaving  congruence  invariant.  This  group,  as 
a  little  reflection  will  show  you,  has  many  sub-groups, 
infinitely    many;     the    set    of   displacements    leaving    a 


214  MATHEMATICAL   PHILOSOPHY 

specified  point  invariant  is  such  a  sub-group;  the  set 
leaving  two  given  points  unchanged  is  another.  How  is 
the  latter  related  to  the  sub-group  leaving  only  one  of  the 
two  points  invariant?  Is  there  a  displacement  leaving 
three  non-collinear  points  invariant?  Do  the  displace- 
ments leaving  a  line  unchanged  constitute  a  group?  Such 
questions  are  but  samples  of  many  that  you  will  find  it 
profitable  to  ask  and  to  try  to  answer. 

For  the  sake  of  emphasis,  permit  me  to  repeat  the  two 
big  questions:  (i)  Given  a  group  of  transformations, 
what  things  are  unchanged  by  them?  (2)  Given  some- 
thing— an  object  or  property  or  relation,  no  matter  what 
— that  is  to  remain  invariant,  what  are  the  groups  of 
transformations,  and  especially  the  largest  group,  that 
leave  the  thing  unaltered?  You  may  wish  to  say:  I 
grant  that  the  questions  are  interesting,  and  I  do  not 
deny  that  they  are  big — big  in  the  sense  of  giving  rise  to 
innumerable  problems  and  big  in  the  sense  that  many  of 
the  problems  are  difficult;  but  I  do  not  see  that  they  are 
big  with  importance.  Why  should  I  bother  with  them? 
In  reply  I  shall  not  undertake  to  say  why  you  should 
bother  with  them;  it  is  sufficient  to  remind  you  that  as 
human  beings  you  cannot  help  it  and  you  do  not  desire  to 
do  so.  In  the  preceding  lecture,  we  saw  that  the  sovereign 
impulse  of  Man  is  to  find  the  answer  to  the  question: 
what  abides?  We  saw  that  Thought, — taken  in  the  widest 
sense  to  embrace  art,  philosophy,  religion,  science,  taken 
in  their  widest  sense, — is  the  quest  of  invariance  in  a 
fluctuant  world.  We  saw  that  the  craving  and  search  for 
things  eternal  is  the  central  binding  thread  of  human 
history.  We  saw  that  the  passion  for  abiding  reality  is 
itself  the  supreme  invariant  in  the  life  of  reason.  And 
we  saw  that  the  bearings  of  the  mathematical  theory 


THE   GROUP  CONCEPT  215 

of  invariance  upon  the  universal  enterprise  of  Thought 
are  the  bearings  of  a  prototype  and  guide.  It  is  evident 
that  the  same  is  true  of  the  mathematical  theory  of 
groups.  Our  human  question  is:  what  abides?  As 
students  of  thought  and  the  history  of  thought,  we  have 
learned  at  length  that  the  question  can  not  be  answered 
fully  at  once  but  only  step  by  step  in  an  endless  progres- 
sion. And  now  what  are  the  steps?  You  can  scarcely 
fail  to  see,  if  you  reflect  a  little,  that  each  of  them, — 
whether  taken  by  art  or  by  science  or  by  philosophy, — 
consists  virtually  in  ascertaining  either  the  invariants 
under  some  group  of  transformations  or  else  the  groups 
of  transformations  that  leave  some  thing  or  things 
unchanged. 

Groups  as  Instruments  for  Defining,  Delimiting,  Dis- 
criminating and  Classifying  Doctrines. — The  foregoing 
question  (2)  has  another  aspect,  which  I  believe  to  be  of 
profound  interest  to  all  students  except  those,  if  there  be 
such,  who  are  insensate  to  things  philosophical.  I  mean 
that,  if  and  whenever  you  ascertain  the  group  of  all  the 
transformations  that  leave  invariant  some  specified  object 
or  objects  of  thought,  you  thereby  define  perfectly  some 
actual  (or  potential)  branch  of  science— some  actual  (or 
potential)  doctrine.  I  will  endeavor  to  make  this  fact 
evident  by  a  few  simple  examples,  and  I  will  choose  them 
from  the  general  field  of  geometry,  though,  as  you  will 
perceive,  such  examples  might  be  taken  from  other  fields. 

For  a  first  example,  consider  the  above-mentioned 
group  D  of  the  displacements  of  our  space.  I  say  that 
this  group  defines  a  geometry  of  space,  which  may  be 
called  the  geometry  of  displacements.  It  defines  it  by 
defining,  or  delimiting,  its  subject-matter.  What  is  its 
subject-matter?     What  does  the  geometry  study?     The 


216  MATHEMATICAL   PHILOSOPHY 

two  questions  are  not  equivalent.  It  studies  all  the 
figures  in  space  but  it  does  not  study  all  their  properties. 
Its  subject-matter  consists  of  those  properties  which  it 
does  study.  What  are  these?  They  are  those  and  only 
those  properties  (of  figures)  that  remain  invariant  under 
all  displacements  but  under  no  other  transformations  of 
space.  The  geometry  of  displacements  might  be  called 
congruence  geometry.  It  includes  the  greater  part  of  the 
ordinary  geometry  of  high  school  but  not  all  of  it,  for  the 
latter  deals,  for  example,  with  similarity  of  figures; 
similarity  is  indeed  invariant  under  displacements,  but 
it  is  also  invariant  under  other  transformations — the 
so-called  similitude  transformations,  to  be  mentioned 
presently. 

For  a  second  example,  consider  the  following.  I  may 
wish  to  confine  my  study  of  spatial  figures  to  their  shape. 
The  doctrine  thus  arising  may  be  called  the  geometry  of 
shape,  or  shape  geometry.  If  I  tell  you  that  I  am  study- 
ing shape  geometry  and  you  ask  me  what  I  mean  by  the 
geometry  of  shape,  there  are  two  ways  in  which  I  may 
answer  your  question.  One  of  the  ways  requires  me  to 
define  the  term  shape — shape  of  a  geometric  figure;  the 
other  way, — the  group  way, — does  not.  Let  us  examine 
them  a  little.  I  have  never  seen  a  mathematical  definition 
of  shape,  but  it  may,  I  believe,  be  precisely  defined  as 
follows.  We  must  distinguish  the  three  things:  sameness 
of  shape;  shape  of  a  given  figure;  and  shape  of  a  figure. 
I  will  define  the  first;  then  the  second  in  terms  of  the 
first;  and,  finally,  the  third  in  terms  of  the  second. 
Two  figures  F  and  F'  will  be  said  to  have  the  same  shape 
if  and  only  if  it  is  possible  to  set  up  a  one-to-one  corre- 
spondence between  the  points  of  F  and  those  of  Ff,  such 
that,  AB  and  CD  being  any  distances  between  points  of 


THE   GROUP  CONCEPT  217 

Fy  and  A'B'  and  CD'  being  the  distances  between  the 
corresponding  points  of  F',  AB/CD=A'B' '/CD'.  Two 
figures  having  the  same  shape  will  be  said  to  be  similar, 
and  conversely.  Having  defined  sameness  of  shape,  or 
similarity,  of  figures,  I  will  define  the  term  "  shape  of  a  - 
given  figure  "  as  follows :  if  F  be  a  given,  or  specific,  figure, 
the  shape  of  F  is  the  class  <r  of  all  figures  similar  to  F;  it 
is  evident  that,  if  F  and  F'  are  not  similar,  the  class  a 
and  the  class  a' — the  shape  of  F' — have  no  figures  in 
common;  it  is  evident,  moreover,  that  there  are  as  many 
o-'s  as  there  are  figure  shapes.  And  now  what  do  we 
mean  by  the  general  term  shape,  or — what  is  tantamount 
— shape  of  a  figure?  What  the  answer  must  be  is  pretty 
obvious:  shape  is  the  class  2  of  all  the  a's.  Note  that 
S  is  a  class  of  classes  and  that  any  a  is  a  class  of  (similar) 
figures.  Having  defined  the  general  term  shape,  I  have, 
you  see,  virtually  answered  your  question:  what  is  the 
geometry  of  shape  ? 

Let  us  now  see  how  the  question  may  be  answered  by 
means  of  the  group  concept.  Two  congruent  figures  are 
clearly  similar,  and  so  similarity  is  invariant  under  the 
group  of  displacements.  But  you  readily  see  that  there 
are  many  other  transformations  under  which  similarity  is 
invariant.  Let  0  be  a  point;  consider  the  bundle  of 
straight  lines, — all  the  lines  through  0, — having  0  for  its 
vertex;  every  point  of  space  is  on  some  line  of  the  bundle; 
let  k  be  any  real  number  (except  zero);  let  P  be  any 
point  and  let  P'  be  such  a  point  on  the  line  OP  that  the 
segment  OP'  =kx  segment  OP;  you  see  that  each  point  P 
is  transformed  into  a  point  P';  the  transformation  is 
called  homothetic;  its  effect,  if  k  be  positive  and  exceed  I, 
is  a  uniform  expansion  of  space  from  0  outward  in  all 
directions;    if  k  be  positive  and  less  than  I,  the  effect  is 


218  MATHEMATICAL   PHILOSOPHY 

contraction  toward  0\  if  k  be  negative,  the  effect  is  such 
an  expansion  or  contraction,  followed  or  preceded  by 
reflection  in  0  as  in  a  mirror;  distances  are  clearly  not 
preserved,  but  distance  ratios  are;  that  is,  if  A,  B,  C 
be  any  three  points  and  if  their  respective  transforms 
under  some  nomothetic  transformation  be  Af,  B',  Cf,  it 
is  evident  that  AB/BC  =  A'B' '/B'C;  accordingly,  if  F 
be  the  transform  of  a  figure  F,  the  figures  are  similar, — 
they  have  the  same  shape  but  not  the  same  size, — they 
are  not  congruent:  similarity  is,  then,  invariant  under  all 
homothetic  transformations,  and  hence  under  combina- 
tions of  them  with  one  another  and  with  displacements; 
the  displacements  and  the  homothetic  transformations 
together  with  all  such  combinations  constitute  a  group 
called  the  group  of  similitude  transformations ;  it  contains 
all  and  only  such  space  transformations  as  leave  similarity 
unchanged.  Here,  then,  is  our  group  definition  of  shape 
geometry:  namely,  the  geometry  of  shape  is  the  study  of 
that  property  of  figures  which  is  common  to  every  figure 
and  its  transforms  under  each  and  all  transformations  of  the 
similitude  group.  Observe  that  this  definition,  unlike 
the  former  one,  employs  neither  the  notion  of  shape  in 
general,  nor  that  of  the  shape  of  a  given  figure;  it  employs 
only  the  notion  of  similarity — sameness  of  shape. 

We  ought,  I  think,  to  consider  one  more  example  of 
how  a  group  of  transformations  serves  to  determine  the 
nature  and  limits  of  a  doctrine  and  thereby  to  discriminate 
the  doctrine  from  all  others.  I  will  again  take  a  geometric 
example,  but  for  the  sake  of  simplicity  I  will  choose  it 
from  the  geometry  of  the  plane  (instead  of  space) .  Before 
presenting  it,  let  me  adduce  a  yet  simpler  example  of  the 
same  kind  taken  from  the  geometry  of  points  in  a  straight 
line.     In  Lecture  IV,  I  explained  what  is  meant  by  a 


THE   GROUP  CONCEPT  219 

projective  line — an  ordinary  line  endowed  with  an 
"  ideal  "  point,  or  point  at  infinity,  where  the  line  meets 
all  lines  parallel  to  it.  Let  I  be  a  projective  line.  In 
the  preceding  lecture,  we  gained  some  acquaintance  with 
the  transformations  of  the  form 

,     ax  +  b 
cx-\-d 

where  the  coefficients  are  any  real  numbers  such  that 
ad  —  be 9^0;  we  saw  that  there  are  00 3  such  transformations 
and  that  each  of  them  converts  the  points  of  L  into  the 
points  of  L  in  such  a  way  that  the  anharmonic  ratio  of 
any  four  points  is  equal  to  the  anharmonic  ratio  of  their 
transforms.  Distances  are  not  preserved;  neither  are  the 
ordinary  ratios  of  distances  preserved;  hence  neither 
congruence  nor  similarity  is  invariant;  no  relation  among 
points — that  is,  no  property  of  figures  (for  here  a  figure  is 
simply  a  set  of  points  on  L) — is  invariant  except  anhar- 
monic ratio  and  properties  expressible  in  terms  of  the 
latter;  no  other  transformations  leave  these  properties 
invariant.  By  a  little  finger  work  you  can  prove  in  a 
formal  way  that  these  transformations  constitute  a  group. 
I  will  merely  indicate  the  procedure,  leaving  it  to  you  to 
carry  it  out  if  you  desire  to  do  so.  The  transformations 
differ  only  in  their  coefficients.  Let  {au  b\,  d,  d\), 
(<22,  #2,  C2,  ^2),  (as,  &3,  C3,  ds)  be  any  three  of  the  trans- 
formations; consider  the  first  and  second;  the  rule  o 
of  combination  is  to  be:  operate  with  the  first  and  then 
on  the  result  with  the  second.  The  first  converts  point 
x  into  point  x': 

a\x+bi 
(2)  x'=  ; 

C\x-\-d\ 


220  MATHEMATICAL   PHILOSOPHY 

the  second  converts  x'  into  point  x" : 

a2x'  -\-b2 


(3)  *"  = 


c2x'  -\-d2 


in  (3)  replace  x'  by  its  value  given  by  (2),  simplify  and  then 
notice  that  you  have  a  transformation  of  form  (1)  con- 
verting x  directly  into  x" .  This  shows  that  the  set  of 
transformations  have  the  group  property.  To  show  that 
they  obey  the  associative  law,  it  is  sufficient  to  perform 
the  operations 

(4)  Oi,  bu  d,  di)o[(a2,  b2,  c2,  d2)o(a3,  h,  c3,  d3)], 

(5)  l(*i>  bi,  ci,  di)o{a2,  b2,  c2,  d2)]o(a3,  b3,  c3,  ds), 

and  then  to  observe  that  the  results  are  the  same.  The 
identical  element  i  is  (a,  o,  o,  a)—  that  is,  the  transforma- 
tion, x'  =x.  The  inverse  of  any  transformation  (a,  b,  c,  d) 
is  (—  d,  by  c,  —a)  for  you  can  readily  show  that  combina- 
tion of  these  gives  (a,  0,  0,  a). 

The  fact  to  be  specially  noted  is  that  this  group  of 
so-called  homographic  transformations  defines  a  certain 
kind  of  geometry  in  the  line  L — namely,  its  projective 
geometry.  In  a  line  there  are  various  geometries;  among 
these  the  projective  geometry  is  characterized  by  its  sub- 
ject-matter, and  its  subject-matter  consists  of  such  proper- 
ties of  point  sets,  or  figures,  as  remain  invariant  under  its 
homographic  group. 

And  now  I  come  to  the  example  alluded  to  a  moment 
ago — the  one  to  be  taken  from  geometries  in  (or  of)  a 
plane.  The  foregoing  homographic  group — in  a  line,  a 
one-dimensional  space — has  an  analogue  in  a  projective 
plane,  another  in  ordinary  3-dimensional  projective  space, 
another  in  a  projective  space  of  four  dimensions,  and  so  on 


THE   GROUP  CONCEPT 


U\ 


ad  infinitum.  What  is  the  analogous  group  for  a  plane  ? 
In  a  chosen  plane  take  a  pair  of  axes  as  explained  in 
Lecture  V,  and  consider  the  pair  of  equations 


(i) 


y  = 


Ax+By  +  C 
Gx  +  Hy  +  I 

Dx+Ey+F 


Gx+Hy+I 

where  the  coefficients  are  any  real  numbers  such  that 

ABC 
(i')      D  E    F    ^0; 
GUI 
i.e.,  such  that 

(i')  AEI-CEG+CDII-BDI+BFG-AFH^O. 

The  coefficients  furnish  eight  independent  ratios,— called 
"parameters," — and  so  we  have  oo  s  equation  pairs  of  form 
(i);  choose  any  one  of  them  and  notice  that  it  is  a  trans- 
formation converting  the  number  pair  (x,  y)  into  a  number 
pair  (x'y  y')y  and  so  converting  the  point  (x,  y)  into  a 
point  (V,  y')\  owing  to  the  inequality  (i')>  any  point 
(x,  y)  is  transformed  into  a  definite  point  (x',y').  In 
any  line  ax'  +by'  +c  =  Q,  replace  x'  and  y'  by  their  values 
given  by  (i),  and  simplify;  the  resulting  equation  is  of 
first  degree  in  x  and  y  and  hence  represents  a  line;  hence, 
you  see,  points  of  a  straight  line  are  converted  into  points 
of  a  straight  line — the  relation,  collinearity,  is  preserved; 
so  is  copunctality — a  pencil  of  lines  has  a  pencil  for  its 
transform;  you  can  readily  show  that  order  is  not  pre- 
served, nor  distances  nor  ordinary  distance-ratios,  nor 
angles;  hence,  if  the  figure  F'  be  the  transform  of  F,  the 
two  figures  are,  in  general,  neither  congruent  nor  similar; 


222  MATHEMATICAL    PHILOSOPHY 

we  say,  however,  that  F  and  F'  are  projective  because, 
as  can  be  proved,  the  anharmonic  ratio  of  any  4  points 
(or  lines)  of  either  is  equal  to  that  of  the  corresponding 
(transform)  points  (or  lines)  of  the  other.  By  the  method 
indicated  for  the  homographic  transformations  of  a  line, 
you  can  prove  that  the  00 8  transformations  of  form  (1) 
constitute  a  group. 

Just  as  a  point  of  the  plane  has  two  coordinates  (x,  y), 
so  a  line  depends  on  two  coordinates;  there  are  various 
ways  to  see  that  such  is  the  case;  an  easy  way  is  this: 
the  line,  ax-\-by-\-c  =  0,  depends  solely  upon  the  ratios 
{a  :  b  :  c)  of  the  coefficients;  these  three  ratios  are  not 
independent — two  of  them  determine  the  third  one;  you 
thus  see  that  the  line  depends  upon  only  two  independent 
variables — it  has,  like  the  point,  two  coordinates;  let  us 
denote  them  by  (u,  v).     Now  consider  the  transformations 


(2) 


,  _Ju-\-Kv+L 
U  ~  Pu+Qv+R* 
Mu+Nv+O 
Pu+Qv+R  ' 


v  — 


where  the  coefficients  are  subject  to  a  relation  like  (1'). 
We  saw  that  a  transformation  (1)  converts  points  into 
points  directly  and  lines  into  lines  indirectly;  just  so,  a 
transformation  (2)  converts  lines  into  lines  directly  and 
points  into  points  indirectly;  hence  the  group  of  line-to- 
line  transformations  (2)  is  essentially  the  same  as  the 
foregoing  group  of  point-to-point  transformations  (1). 
This  latter  group  is  called  the  group  of  collineations  of 
(or  in)  the  plane. 

I  am  going  now  to  ask  you  to  notice  an  ensemble  of 
transformations  (of  the  plane)  that  are  neither  point-to- 


THE  GROUP  CONCEPT  223 

point  nor  line-to-line  transformations  but  are  at  once 
point-to-line  and  line-to-point  transformations.  These 
are  represented  by  the  pair  of  formulas 


(3) 


ax-\-by-\-c 

u= /■ . 

gx+hy+i 

dx-\~ey-\-f 
gx-\-hy-\-i 


v  = 


where  the  coefficients  are  again  subject  to  a  relation  like 
(i').  Any  such  transformation  converts  a  point  (x,  y) 
into  a  line  {u,  v);  now  operate  on  the  points  of  this  line 
by  the  same  transformation  or  another  one  of  form  (3); 
the  points  are  converted  into  lines  constituting  a  pencil 
having  a  vertex,  say  (xf,  y');  thus  the  combination  con- 
verts point  (x,  y)  into  point  (xr,  y') — it  is  a  point-to-point 
transformation  and  hence  belongs  to  the  group  of  collinea- 
tions;  you  thus  see  that  the  set  of  transformations  (3) 
is  not  a  group;  but  this  set  and  the  collineations  together 
constitute  a  group  including  the  collineations  as  a  sub- 
group. This  large  group  is  called  the  Group  of  Projective 
Transformations  of  the  Plane.  Why?  Because  every 
transformation  in  it  and  no  other  transformation  leaves 
all  anharmonic  ratios  unchanged. 

What  is  the  projective  geometry  of  the  plane?  The 
group  now  in  hand  enables  us  to  answer  the  question 
perfectly.  The  answer  is:  Projective  plane  geometry  is 
that  geometry  which  studies  such  and  only  such  properties 
of  plane  figures  as  remain  invariant  under  the  group  of 
projective  transformations. 

In  reading  the  essays  of  the  late  Henri  Poincar^  you 
have  met  the  statement:  "  Euclidean  space  is  simply  a 
group."     The  foregoing  examples  should  enable  you  to 


224  MATHEMATICAL   PHILOSOPHY 

understand  its  meaning.  And  they  should  lead  you  to 
surmise — what  is  true — that  answers  like  the  foregoing 
ones  are  available  for  similar  questions  regarding  all 
geometries  of  a  space  of  any  number  of  dimensions  and 
— what  is  more — regarding  mathematical  doctrines  in 
general.  Whatsoever  things  are  invariant  under  all  and 
only  the  transformations  of  some  group  constitute  the 
peculiar  subject-matter  of  some  (actual  or  potential) 
branch  of  knowledge.  And  you  see  that  every  such  group- 
defined  science  views  its  subject  matter  under  the  aspect  of 
eternity. 

A  Question  for  Psychologists. — Before  closing  this 
lecture,  I  desire  to  speak  briefly  of  two  additional  matters 
connected  with  the  notion  of  group:  one  of  the  matters 
is  psychological:  the  other  is  historical.  Being  students 
of  philosophy,  you  are  obliged  to  have  at  least  a  good 
secondary  interest  in  psychology.  I  wish  to  propose  for 
your  future  consideration  a  psychological  question — one 
which  psychologists  (I  believe)  have  not  considered  and 
which,  though  it  has  haunted  me  a  good  deal  from  time 
to  time  in  recent  years,  I  am  not  yet  prepared  to  answer 
confidently.  The  question  is:  Is  mind  a  group?  Let  us 
restrict  the  question  and  ask:  Is  mind  a  closed  system — 
that  is,  has  it  the  group  property  ?  Some  of  the  difficulties 
are  immediately  obvious.  In  order  that  the  question 
shall  have  definite  meaning,  it  is  necessary  to  think  of 
mind  as  a  system  composed  of  a  class  of  things  and  a  rule, 
or  law,  of  combination  by  which  each  of  the  things  can  be 
combined  with  itself  and  with  each  of  the  other  things. 
We  may  make  a  beginning  by  saying  that  the  required 
class  is  the  class  of  mental  phenomena.  But  what  does 
the  class  include?  What  phenomena  are  members  of  it? 
Some  phenomena, — feeling,  for  example,  seeing,  hearing, 


THE  GROUP  CONCEPT  225 

tasting,  thinking,  believing,  doubting,  craving,  hoping, 
expecting, — are  undoubtedly  mental;  others  seem  not  to 
be — as  what  I  see,  for  example,  what  I  taste,  what  I 
believe,  and  the  like.  Here  are  difficulties.  You  will 
find  a  fresh  and  suggestive  treatment  of  them  in  Bertrand 
Russell's  The  Ultimate  Constituents  of  Matter,  found  in  the 
author's  Mysticism  and  Logic  and  especially  in  his 
Analysis  of  Mind.  Let  us  suppose  that,  despite  the  dif- 
ficulties in  the  way,  you  have  decided  what  you  are  going 
to  call  mental  phenomena.  You  have  then  to  consider 
the  question  of  their  combination.  We  do  habitually 
speak  of  combining  mental  things:  hoping,  for  example, 
is,  in  some  sense,  a  union,  or  combination,  of  desiring  and 
expecting;  the  feeling  called  patriotism  is  evidently  a 
combination  of  a  pretty  large  variety  of  feelings;  in  the 
realm  of  ideas, — which  you  will  probably  desire  to  include 
among  mental  phenomena, — we  have  seen  that,  for 
example,  the  idea  named  "  vector  "  is  a  union,  or  combina- 
tion, of  the  ideas  of  direction,  sense  and  length;  and  so  on 
— examples  abound.  But  does  combination  as  a  process 
or  operation  have  the  same  meaning  in  all  such  cases? 
It  seems  not.  What,  then,  is  your  rule  o  to  be?  Possibly 
the  difficulty  could  be  surmounted  as  follows:  if  you 
discovered  that  some  mental  phenomena  are  combinable 
by  a  rule  o\,  others  by  a  rule  02,  still  others  by  a  rule  03, 
and  so  on,  thus  getting  a  finite  number  of  particular  rules, 
you  could  then  take  for  a  more  general  rule  0  the  disjunc- 
tion, or  so-called  logical  sum,  of  the  particular  ones;  that 
is,  you  could  say  that  rule  0  is  to  be :  0 1  or  02  or  03  or  . .  .  or  o„; 
so  that  two  phenomena  would  be  combinable  by  0  if  they 
were  combinable  by  one  or  more  of  the  rules  o\,  o2,  .  .  .  ,  on. 
If  you  thus  found  a  rule  by  which  every  two  of  the  phe- 
nomena you  had  decided  to  call  mental  admitted  of  com- 


226  MATHEMATICAL    PHILOSOPHY 

bination,  then  your  final  question  would  be:  is  the  result 
of  every  such  combination  a  mental  phenomenon?  That 
is  not  quite  the  question;  for  under  the  rule  two  phe- 
nomena might  be  combinable  in  two  or  more  ways,  and 
some  of  the  results  might  (conceivably)  be  mental  and  the 
others  not;  so  your  question  would  be:  can  every  two 
mental  phenomena  be  combined  under  your  rule  so  as  to 
yield  a  mental  phenomenon?  If  so,  then  mind,  as  you  had 
defined  it,  would  have  the  group  property  under  some 
rule  of  combination.  If  you  found  mind  to  have  the 
group  property  under  some  rule  or  rules  but  not  under 
others,  you  would  be  at  once  confronted  with  a  further 
problem,  which  I  will  not  tarry  to  state. 

We  have  been  speaking  of  mind — of  mind  in  general. 
Similar  questions, — perhaps  easier  if  not  more  fruitful 
questions, — can  be  put  respecting  particular  minds — 
your  mind,  mine,  John  Smith's.  Has  every  individual 
mind  the  group  property?  Has  no  such  mind  the  prop- 
erty? Have  some  of  them  the  property  and  others 
not? 

It  seems  very  probable  that  the  answer  to  the  first  of 
the  questions  must  be  negative.  There  are  at  all  events 
some  minds  having  (presenting,  containing)  mental 
phenomena  that  are  definitely  combinable  in  a  way  to 
yield  mental  phenomena  that  nevertheless  do  not  belong 
to  those  minds.  What  is  meant  is  this:  a  given  mind 
may  possess  certain  ideas  which  are  combinable  so  as  to 
form  another  idea;  it  may  happen  that  the  mind  in  ques- 
tion is  incapable  of  grasping  the  new  idea.  Such  minds 
have  no  doubt  come  under  the  observation  of  every 
experienced  teacher.  I  myself  have  seen  many  such  cases 
and  remember  one  of  them  very  vividly:  that  of  a  young 
woman  who  had  made  a  brilliant  record  in  undergraduate 


THE   GROUP  CONCEPT  227 

collegiate  mathematics  including  the  elements  of  analyt- 
ical geometry  and  calculus;  and  who,  encouraged  by  this 
initial  success,  aspired  to  the  mathematical  doctorate  and 
entered  seriously  upon  higher  studies  essential  thereto; 
it  was  necessary  for  her  to  grasp  more  and  more  complicate 
concepts  formed  by  combining  ideas  she  already  pos- 
sessed; after  no  long  time  she  reached  the  limit  of  her 
ability  in  this  matter, — a  fact  first  noticed  by  her  instruc- 
tors and  then  by  herself, — and  being  a  woman  of  good 
sense,  she  abandoned  the  pursuit  of  higher  mathematics. 
I  may  add  that  subsequently  she  gained  the  doctorate  in 
history.  It  may  be  that  some  minds  are  not  thus  limited. 
It  may  be  that  a  genius  of  the  so-called  universal  type, — 
an  Aristotle,  for  example,  or  a  Leibniz  or  a  Leonardo 
da  Vinci, — is  one  whose  mind  has  the  group  property. 
May  I  leave  the  questions  for  your  consideration  in  the 
days  to  come? 

The  Group  Concept  Dimly  Adumbrated  in  Early 
Philosophic  Speculation. — The  mathematical  theory  of 
groups  is  immense  and  manifold;  in  the  main  it  is  a  work 
of  the  last  sixty  years;  even  the  germ  of  it  seems  not  to 
antedate  Ruffini  and  Lagrange.  Why  so  modern  ?  Why 
did  not  the  concept  of  a  closed  system, — of  a  system  having 
the  group  property,— come  to  birth  many  centuries  earlier? 
The  elemental  constituents  of  the  concept, — the  idea  of  a 
class  of  things,  the  idea  of  anything  being  or  not  being  a 
member  of  a  class,  the  idea  of  a  rule  or  law  of  combination, 
— all  these  were  as  familiar  thousands  of  years  ago  as  they 
are  now.  The  question  is  one  of  a  host  of  similar  ques- 
tions whose  answers,  if  ever  they  be  found,  will  constitute 
what  in  a  previous  lecture  I  called  the  yet  unwritten 
history  of  the  development  of  intellectual  curiosity.  Who 
will  write  that  history?     And  when? 


228  MATHEMATICAL   PHILOSOPHY 

The  fact  that  the  precise  formation  of  the  mathematical 
concept  of  group  is  of  so  recent  date  is  all  the  more  curious 
because  an  idea  closely  resembling  that  of  group  has 
haunted  the  minds  of  a  long  line  of  thinkers  and  is  found 
stalking  like  a  ghost  in  the  mist  of  philosophic  speculation 
from  remote  antiquity  down  even  to  Herbert  Spencer. 
I  refer  to  those  worldwide,  age-long,  philosophic  specula- 
tions which,  because  of  their  peculiar  views  of  the  uni- 
verse, may  be  fitly  called  the  Philosophy  of  the  Cosmic 
Cycle  or  Cosmic  Year.  This  philosophy,  despite  the  spell 
of  a  certain  beauty  inherent  in  it,  has  lost  its  vogue. 
To-day  we  are  accustomed  to  thinking  of  the  universe  as 
undergoing  a  beginningless  and  endless  evolution  in  course 
of  which  no  aspect  or  event  ever  was  or  ever  will  be 
exactly  repeated.  In  sharpest  contrast  with  that  con- 
ception, the  philosophy  of  the  cosmic  cycle  regards  all 
the  changes  of  which  the  universe  is  capable  as  constitut- 
ing an  immense  indeed  but  finite  and  closed  system  of 
transformations,  which  follow  each  other  in  definite  suc- 
cession, like  the  spokes  of  a  gigantic  revolving  wheel, 
until  all  possible  changes  have  occurred  in  the  lapse  of  a 
long  but  finite  period  of  time — called  a  cosmic  cycle  or 
cosmic  year — whereupon  everything  is  repeated  precisely, 
and  so  on  and  on  without  end.  This  philosophy,  I  have 
said,  has  lost  its  vogue;  but,  if  the  philosophy  be  true, 
it  will  regain  it,  for,  if  true,  it  belongs  to  the  cosmic 
cycle  and  hence  will  recur.  The  history  of  the  philosophy 
of  the  cosmic  year  is  exceedingly  interesting  and  it  would, 
I  believe,  be  an  excellent  subject  for  a  doctor's  disserta- 
tion. The  literature  is  wide-ranging  in  kind,  in  place  and 
in  time.  Let  me  cite  a  little  of  it  as  showing  how  closely 
its  central  idea  resembles  the  mathematical  concept  of  a 
cyclic  group. 


THE  GROUP  CONCEPT  229 

In  his  Philosophie  der  Griechen  (Vol.  Ill,  2nd  edition) 
Zeller,  speaking  of  the  speculations  of  the  Stoics,  says: 

Out  of  the  original  substance  the  separate  things  are 
developed  according  to  an  inner  law.  For  inasmuch  as 
the  first  principle,  according  to  its  definition,  is  the 
creative  and  formative  power,  the  whole  universe  must 
grow  out  of  it  with  the  same  necessity  as  the  animal 
or  the  plant  from  the  seed.  The  original  fire,  according 
to  the  Stoics  and  Heraclitus,  first  changes  to  "  air  " 
or  vapor,  then  to  water;  out  of  this  a  portion  is  pre- 
cipitated as  earth,  another  remains  water,  a  third 
evaporates  as  atmospheric  air,  which  again  kindles  the 
fire,  and  out  of  the  changing  mixture  of  these  four 
elements  there  is  formed, — from  the  earth  as  center, — 
the  world.  .  .  .  Through  this  separation  of  the  elements 
there  arises  the  contrast  of  the  active  and  the  passive 
principle:  the  soul  of  the  world  and  its  body.  .  .  .  But 
as  this  contrast  came  in  time,  so  it  is  destined  to  cease; 
the  original  substance  gradually  consumes  the  matter, 
which  is  segregated  out  of  itself  as  its  body,  till  at  the 
end  of  this  world-period  a  universal  world  conflagration 
brings  everything  back  to  the  primeval  condition.  .  .  . 
But  when  everything  has  thus  returned  to  the  original 
unity,  and  the  great  world-year  has  run  out,  the  forma- 
tion of  a  new  world  begins  again,  which  is  so  exactly 
like  the  former  one  that  in  it  all  things,  persons  and 
phenomena,  return  exactly  as  before;  and  in  this  wise 
the  history  of  the  world  and  the  deity  .  .  .  moves  in  an 
endless  cycle  through  the  same  stages. 

A  similar  view  of  cosmic  history  is  present  in  the 
speculations  of  Empedocles,  for  whom  a  cycle  consists  of 
four  great  periods:  Predominant  Love — a  state  of  com- 
plete aggregation;  decreasing  Love  and  increasing  Hate; 
predominant  Strife — complete  separation  of  the  elements; 
decreasing  Strife  and  increasing  Love.  At  the  end  of  this 
fourth  period,  the  cycle  is  complete  and  is  then  repeated — 


230  MATHEMATICAL   PHILOSOPHY 

the  history  of  the  universe  being  a  continuous  and  end- 
lessly repeated  vaudeville  performance  of  a  single  play. 

Something  like  the  foregoing  seems  to  be  implicit  in 
the  following  statement  by  Aristotle  in  the  Metaphysics: 

Every  art  and  every  kind  of  philosophy  have  prob- 
ably been  invented  many  times  up  to  the  limits  of  what 
is  possible  and  been  again  destroyed. 

And  in  Ecclesiastes  (III,  15): 

That  which  hath  been  is  now;  and  that  which  is 
to  be  hath  already  been. 

Even  Herbert  Spencer  at  the  close  of  his  First  Prin- 
ciples speaks  as  follows: 

Thus  we  are  led  to  the  conclusion  that  the  entire 
process  of  things,  as  displayed  in  the  aggregate  of  the 
visible  universe,  is  analogous  to  the  entire  process  of 
things  as  displayed  in  the  smallest  aggregates.  Motion 
as  well  as  matter  being  fixed  in  quantity,  it  would  seem 
that  the  change  in  the  distribution  of  matter  which 
motion  effects,  coming  to  a  limit  in  whatever  direction 
it  is  carried,  the  indestructible  motion  necessitates  a 
reverse  redistribution.  Apparently  the  universally 
coexistent  forces  of  attraction  and  repulsion,  which 
necessitate  rhythm  in  all  minor  changes  throughout  the 
universe,  also  necessitates  rhythm  in  the  totality  of 
changes — alternate  eras  of  evolution  and  dissolution. 
And  thus  there  is  suggested  the  conception  of  a  past 
during  which  there  have  been  successive  evolutions 
analogous  to  that  which  is  now  going  on;  and  a  future 
during  which  successive  other  evolutions  may  go  on — 
ever  the  same  in  principle  but  never  the  same  in  con- 
crete result. 

Spencer  was,  you  know,  but  poorly  informed  in  the  history 
of  thought  and  he  was  probably  not  aware  that  the  main 


THE  GROUP  CONCEPT  231 

idea  in  the  lines  just  now  quoted  was  ancient  two  thousand 
years  before  he  was  born.  You  should  note  that  the 
Spencerian  universe  of  transformations  narrowly  escapes 
being  a  closed  system — escapes  by  the  last  six  words  of 
foregoing  quotation.  The  cosmic  cycles  do  indeed  follow 
each  other  in  an  endless  sequence — "  ever  the  same  in 
principle  but  never  the  same  in  concrete  result."  The 
repetitions  are  such  "  in  principle  "  only,  not  in  result — 
there  is  always  something  new. 

One  of  the  very  greatest  works  of  man  is  the  De  Rerum 
Natura  of  Lucretius — immortal  exposition  of  the  thought 
of  Epicurus,  "  who  surpassed  in  intellect  the  race  of  man 
and  quenched  the  light  of  all  as  the  ethereal  sun  arisen 
quenches  the  stars."  Neither  a  student  of  philosophy  nor 
a  student  of  natural  science  can  afford  to  neglect  the  read- 
ing of  that  book.  For,  although  it  contains  many,— very, 
very  many, — errors  of  detail, — some  of  them  astonishing 
to  a  modern  reader, — yet  there  are  at  least  four  great 
respects  in  which  it  is  unsurpassed  among  the  works  that 
have  come  down  from  what  we  humans,  in  our  ignorance 
of  man's  real  antiquity,  have  been  wont  to  call  the 
ancient  world:  it  is  unsurpassed,  I  mean,  in  scientific 
spirit;  in  the  union  of  that  spirit  with  literary  excellence; 
in  the  magnificence  of  its  enterprise;  and  in  its  anticipa- 
tion of  concepts  among  the  most  fruitful  of  modern 
science.  For  such  as  can  not  read  it  in  the  original  there 
are,  happily,  two  excellent  English  translations  of  it — one 
by  H.  A.  J.  Munro  and  a  later  one  by  Cyril  Bailey.  Of 
this  work  I  hope  to  speak  further  in  a  subsequent  lecture 
of  this  course.  My  purpose  in  citing  it  here  is  to  signalize 
it  as  being  perhaps  the  weightiest  of  all  contributions  to 
what  I  have  called  the  philosophy  of  the  cosmic  year. 
The  Lucretian  universe  though  not  a  finite  system,  is 


2S2  MATHEMATICAL    PHILOSOPHY 

indeed  a  closed  system,  of  transformations:  any  event, 
whether  great  or  small,  that  has  occurred  in  course  of  the 
beginningless  past  has  occurred  infinitely  many  times  and 
will  recur  infinitely  many  times  in  course  of  an  unending 
future;  and  nothing  can  occur  that  has  not  occurred, — 
there  never  has  been  and  there  never  will  be  aught  that 
is  new, — every  occurrence  is  a  recurrence.  Let  me  say 
parenthetically,  in  passing,  that  such  a  concept  of  the 
universe  is  damnably  depressing  but  not  more  so  than  the 
regnant  mechanistic  hypothesis  of  modern  natural  science. 
In  relation  to  this  hypothesis  you  should  by  no  means 
fail  to  read  and  digest  Professor  W.  B.  Smith's  great 
address:  "Push  or  Pull?"  (Monist,  Vol.  XXIII,  1913). 
See  also  Smith's  "Are  Motions  Emotions?"  (Tulane 
Graduates1  Magazine,  Jan.,  1914).  And  you  should  read 
J.  S.  Haldane's  Life,  Mechanism  and  Personality. 

Should  you  desire  to  pursue  the  matter  further  either 
with  a  view  to  noting  speculative  adumbrations  of  the 
group  concept  or,  as  I  hope,  with  the  larger  purpose  of 
writing  a  historical  monograph  on  the  philosophy  of  the 
Cosmic  Cycle,  the  following  references  may  be  of  some 
service  as  a  clue. 

"  The  Dream  of  Scipio  "  in  Cicero's  Republic  (Hard- 
ingham's  translation). 

Michael  Foster's  Physiology. 

Lyell's  Principles  of  Geology. 

The  fourth  Eclogue  of  Virgil  (verses  31-36). 

Riickert's  poem  Chidher. 

Moleschott's  Kreislauf  des  Lebens. 

Clifford's  "  The  First  and  Last  Catastrophe  "  in  his 
Lectures  and  Essays. 

Inge's  The  Idea  of  Progress  (being  the  Romanes 
Lecture,  1920). 


LECTURE  XIII 
Variables  and  Limits 

A  GLANCE   AT  THE   SHADOWY   BACKGROUND   OF   SCIENTIFIC 

IDEAS — THE  MEANINGS  OF  VARIABLE  AND  CONSTANT 

RANGES    OF   VARIATION   AND   THE   IDEA   OF   NEIGHBOR- 
HOOD  VARIOUS     DEFINITIONS     OF     LIMIT     CLARIFIED 

BY   SIMPLE   EXAMPLES THE    SCANDAL   OF  A   STARVING 

NURSE   IN  THE   RICHEST   LAND   KNOWN  TO   HISTORY. 

In  the  introductory  lecture,  I  spoke  at  length  on  the 
mathematical  obligations  of  philosophy.  In  preparing  to 
discharge  them  it  is  imperatively  necessary  for  students 
of  philosophy  to  gain  genuine  understanding  of  those 
great  concepts  that  are  as  vital  organs  in  the  body  of 
mathematics,  giving  the  science  not  only  its  life  but  its 
character  and  its  power.  It  is  one  of  the  aims  of  these 
lectures  to  assist  students  primarily  interested  in  philoso- 
phy to  gain  such  an  understanding.  In  pursuance  of  that 
aim  it  is,  I  believe,  essential  to  devote  one  or  two  lectures 
to  the  nature  and  the  significance  of  the  mathematical 
concept  denoted  by  the  term  "  limit."  There  are  in  cur- 
rent use  several  concepts  of  the  term,  but  for  the  present 
we  may  speak  as  if  there  were  only  one.  The  importance 
and  the  power  of  the  concept  have  been  so  long  recognized 
by  mathematicians  that  the  notion  of  limit  and  what  is 
often  called  the  method  of  limits  have  found  their  way 
down  into  text-books  of  algebra,  geometry  and  trigonom- 

233 


234  MATHEMATICAL   PHILOSOPHY 

etry  designed  for  high  schools.  In  high  school  you  prob- 
ably learned  something  of  the  lingo  of  limits;  if  you  really 
there  grasped  the  ideas  involved,  you  were  extraordinary 
pupils  or  were  very  fortunate  in  your  teachers  or  both. 
I  say  this  because  in  collegiate  freshman  classes  I  have 
met  many  students  who  in  their  preparation  for  college 
had  been  exposed  to  the  notion  and  method  of  limits,  and 
I  have  the  impression  that  among  them  there  were  very 
few  or  none  whose  wisdom  in  the  matter  was  appreciably 
more  than  phraseological;  in  the  case  of  most  it  was  even 
less.  The  explanation  is  not  far  to  seek:  the  concept  of 
limit  is  a  subtle  concept  and  the  right  use  of  it  in  mathe- 
matical argumentation  is  a  delicate  process;  these  two 
things  can  not  be  caught,  so  to  speak,  on  the  fly;  they 
require  to  be  reflected  upon  again  and  again;  they  are 
among  the  things  that  require  to  be  pondered;  but  such 
meditation,  such  deliberation  upon  elusive  scientific 
abstractions,  is  one  of  the  things  which  boys  and  girls 
of  the  indicated  age  will  not  do;  it  is  not  their  fault; 
if  they  did  it  they  would  not  be  boys  and  girls;  they 
would  be  seasoned  logicians  and  philosophers.  If  such 
understanding  of  the  nature  and  significance  of  limits 
as  you  may  be  supposed  to  have  acquired  in  high  school 
has  not  been  deepened  and  refined  by  subsequent  study, 
the  intervening  years  have  probably  so  dimmed  your 
impressions  of  the  matter  that  you  are  now  fortunately 
able  to  approach  the  subject  afresh,  bringing  to  bear  upon 
it  critical  power  of  sufficient  maturity. 

In  endeavoring  to  analyze  the  concept  of  limit  it 
becomes  immediately  evident  that  it  involves  the  notion 
of  variable  and  the  notion  of  constant,  together  with  such 
other  notions  as  that  of  variable  and  that  of  constant 
themselves  involve.     What  do  mathematicians  mean  by 


VARIABLES  AND  LIMITS  235 

the  terms  variable  and  constant?  The  great  majority  of 
professional  mathematicians  take  the  meanings  of  these 
terms  mainly  for  granted;  they  do  so  because  they  are  so 
occupied  in  teaching  mathematics  or  in  extending  its 
superstructure  as  to  have  but  little  or  no  interest  in 
"  the  nice  sharp  quillets  of  the  law  "  as  revealed  in  its 
logical  foundations;  in  the  foregoing  lectures  I  have  fol- 
lowed the  actual  practice  of  mathematicians  in  respect  to 
the  two  terms  in  question;  I  have,  that  is,  freely  employed 
the  terms,  not  indeed  quite  unconsciously,  but  without 
apology  and  without  explanation,  believing  that  such  use 
would  lead  to  no  appreciable  misunderstanding.  Now, 
however,  before  attempting  to  give  a  formal  definition  of 
the  great  term  "  limit,"  it  will  be  worth  while,  I  believe, 
to  glance  at  its  shadowy  background — to  reflect  a  little  on 
the  meanings  of  the  yet  greater  notions  upon  which  the 
concept  of  a  limit  depends. 

Meaning  of  the  Terms  Variable  and  Constant. — What  is 
the  mathematical  meaning  of  the  term  variable?  It  is 
natural  to  answer,  and  to  answer  very  confidently,  that  a 
variable  is  something  that  varies  or  changes,  like  the 
position  of  an  object  in  motion,  the  time  of  day,  the 
length  of  a  burning  cigar.  We  are  going  to  see  that  this 
answer,  though  perfectly  natural,  is  entirely  wrong.  In  a 
previous  lecture  I  drew  your  attention  to  the  fact  that 
mathematicians  habitually  employ  highly  figurative 
speech;  especially  that  they  are  constantly  employing 
dynamic  terms  in  describing  static  facts.  In  particular 
I  pointed  out  that  it  is  a  common  practice  of  mathema- 
ticians to  use  the  dynamic  term  transformation, — suggest- 
ing change,  variation,  transmutation, — to  denote  what  is 
in  fact  a  static  thing,  namely,  a  relation — something  that 
is  unchanging,  eternal.     There  is  thus  a  striking  incon- 


Z$6  MATHEMATICAL   PHILOSOPHY 

gruity  between  what  is  said,  or  the  manner  of  saying  it, 
and  what  is  meant.  We  are  going  to  see  that  there  is  a 
like  incongruity  between  the  meaning  of  the  mathematical 
term  "  variable  "  and  the  manner  in  which  mathematicians 
habitually  speak  of  variables.  I  do  not  condemn  their 
manner  of  speech;  I  approve  of  it  because  it  is  stimulating 
and  economical  and  because  it  does  not,  except  in  certain 
very  fundamental  questions,  lead  to  error;  but  such 
incongruity  is  a  thing  which  you  as  philosophers  should 
carefully  notice  in  the  interest  of  clarity  and  critical 
understanding.  Mathematicians  do  indeed  habitually 
speak  of  variables  as  if  the  mathematical  concept  of  a 
variable  were  the  concept  of  something  whose  essential 
nature  is  to  suffer  change;  that  is  to  say,  when  they  use 
some  symbol,  say,  x,  to  denote  what  they  call  a  variable, 
they  familiarly  speak  of  the  variable  x  as  altering  its 
value,  as  increasing  or  decreasing,  as  growing  large  or 
growing  small,  as  approaching  or  not  approaching  this  or 
that,  and  so  on;  yet,  in  spite  of  such  a  way  of  speaking, 
what  they  really  mean  by  the  term  "  variable  "  essentially 
involves  no  idea  of  change  whatever  as  "  change  "  is 
commonly  understood.  This  fact  may,  I  believe,  be  made 
sufficiently  evident. 

To  make  it  evident  let  us,  in  seeking  the  meaning  of  the 
term  '  variable,"  recur  to  the  idea  of  propositional 
function;  for,  although  some  of  the  things  called  variables 
in  the  logical  theory  of  propositional  functions  are  not  so 
called  in  traditional  mathematics,  yet  whatever  is  called  a 
variable  in  mathematics  appears  explicitly  or  implicitly 
as  a  variable  in  some  propositional  function;  for  example, 
the  variable  x  in  such  a  propositional  function  as— a;  is  a 
man,  — has  not  yet  gained  full  recognition  as  a  mathe- 
matical variable;    on  the  other  hand,  the  mathematical 


VARIABLES  AND  LIMITS  237 

variable  x  in  an  equation,  say,  ^x2-\-2x—  9=0,  appears  as 
a  variable  in  a  propositional  function,  for,  as  you  know, 
such  an  equation  is  such  a  function;  and,  for  another 
example,  when  the  mathematician  says,  "  I  will  let  x 
represent  any  point  in  a  certain  line  L,"  thereby  indicating 
that  he  will  use  x  as  a  variable,  he  virtually  (implicitly) 
says,  "  I  will  let  x  represent  any  one  of  the  verifiers  of  the 
propositional  function— #  is  a  point  in  the  line  L." 

Now  let  4>(x)  denote  some  given  propositional 
function  involving  one  and  only  one  of  the  things  called 
variables.  I  am  going  to  speak  of  4>(x);  while  I  am  doing 
so,  you  may  find  it  helpful  to  attach  what  is  said  to  some 
simple  specific  function  such  as  "  x  is  a  man  "  or  "  x2  =4" 
or  "  x  is  a  member  of  this  audience."  Our  function 
<t>(x)  contains,  we  say,  one  variable,  namely,  x\  x,  we  say, 
is  a  symbol;  notice  that,  when  speaking  precisely,  we  do 
not  say  that  the  symbol  denotes  the  variable,  we  say  that 
the  symbol  is  the  variable.  The  question  is:  What  is 
meant  by  saying  that  as  here  used  the  symbol  x  is  a 
variable?  Before  attempting  to  answer,  let  us  reflect 
that  there  are  terms  such  that  if  any  one  of  them  be 
substituted  for  x  in  <f>(x)  the  resulting  statement  is  non- 
sensical,— non-significant, — and  hence  neither  true  nor 
false;  and  that  there  are  other  terms  which,  on  being  thus 
substituted,  yield  significant  statements — that  is,  propo- 
sitions (true  or  false).  You  will  recall  that  terms  of  the 
former  kind, — nonsense-giving  terms, — were  described 
in  a  previous  lecture  as  inadmissible  for  <j>(x)  and  that  the 
latter  kind, — sense-giving  terms, — were  described  as  ad- 
missible terms  for  </>(#)•  Now,  it  is  significant  statements, 
— statements  that  are  true  or  false, — propositions, — and 
only  such  that  we  are  concerned  with  when  using  proposi- 
tional functions  as  instruments  in  research  or  in  exposi- 


238  MATHEMATICAL   PHILOSOPHY 

tion,  and  hence  the  admissible  terms  and  only  these  are 
important,  for  it  is  in  virtue  of  these  and  only  these  that 
a  propositional  function  is,  as  we  have  seen,  a  matrix, 
mould,  or  source,  of  propositions.  If  now  we  observe  that 
the  admissible  terms  for  a  given  function  4>(x)  constitute 
a  type,  or  class,  of  terms,  we  shall  be  prepared  to  answer 
our  question.  The  answer  is:  The  symbol  x  in  a  given 
propositional  function  4>(x)  is  called  a  variable  because  the 
symbol  represents  any  one  of  the  terms  of  the  class  of 
admissible  terms  for  4>(x)  and  represents  nothing  else. 
There  is  nothing  subtler  in  human  speech  and  nothing 
more  important  than  the  phrase  "  any  one  "  as  here  used; 
without  it,  logic,  science,  philosophy,  even  the  discourse  of 
the  workaday  world,  would  be  impossible.  What  does 
the  phrase  mean  ?  It  does  not  admit  of  precise  definition, 
for  it  is  essentially  involved  (explicitly  or  implicitly)  in 
the  very  act  of  definition.  The  only  or  the  best  way  to 
sharpen  our  sense  of  its  meaning  is  to  meditate  upon 
examples  of  its  use.  A  farmer  has  in  his  barn  three 
horses — Black,  Sorrel  and  Gray.  He  says  to  his  servant: 
"John,  fetch  me  a  horse  from  the  barn."  John  asks: 
"Which  one?"  "  Any  one,"  replies  the  farmer.  As  here 
used  the  phrase  "  a  horse  "  is  a  variable  because  it  repre- 
sents "  any  one  "  of  a  certain  class  of  horses;  in  represent- 
ing "  any  one  "  of  the  class,  it  does  not  refer  to  a  particular 
horse,  for  evidently  "  any  one  "  is  not  a  description  or 
designation  of  a  particular  one  of  the  horses;  neither  does 
it  refer  to  all  of  the  horses  conjunctively — Black  and 
Sorrel  and  Gray — John  is  not  to  fetch  them  all;  it  does 
refer  to  each  of  them  disjunctively — John  is  to  fetch  Black 
or  Sorrel  or  Gray — no  matter  which  one.  So  it  is  in  the 
foregoing  definition:  in  representing  "  any  one  "  of  the 
class  of  admissible  terms  for  4>{x),  x  does  not  refer  to  a 


VARIABLES  AND   LIMITS  239 

specific  one  of  the  terms  nor  to  all  of  them  conjunctively; 
it  refers  to  each  of  them  disjunctively.  It  is  essential 
and  now  easy  to  see  clearly  that  no  idea  of  variation  or 
change  is  involved:  <f>(x)  being  given,  it  is  timeless, 
unchanging;  the  class  of  its  admissible  terms  is  timeless, 
unchanging;  "  any  one  "  applied  to  the  terms  of  this  class 
is  timeless,  unchanging;  x's  representation  of  this  "  any 
one  "  is  timeless,  unchanging;  you  thus  see  that  a  given 
mathematical  variable  is  timeless,  unchanging;  and  so 
you  see  that,  when  mathematicians  speak  of  a  variable  as 
undergoing  change,  they  speak  metaphorically.  Such 
speech  is,  I  have  said,  very  convenient  and  stimulating, 
and,  now  that  we  recognize  its  metaphorical  character, 
I  shall  feel  at  liberty  to  employ  it  freely  in  this  discussion. 

A  variable  being  given,  the  class  of  terms  "  any  one  " 
of  which  is  represented  by  it  is  commonly  called  the  range 
of  the  variable;  thus  in  the  case  of  our  4>(x),  the  range  of 
x  is  the  class  of  admissible  terms  for  <j)(x).  The  range  of 
a  variable  may  contain  only  one  term,  as,  for  example, 
when  we  say,  "  Let  x  represent  any  point  common  to  the 
given  intersecting  lines  L  and  L'."  Such  a  variable  is 
called  a  constant;  thus  you  see  that  a  mathematical 
constant,  far  from  being  (as  vulgarly  supposed)  the  oppo- 
site of  a  variable,  is  itself  a  variable.  If  a  variable's 
range  be  a  null  class  (an  empty  class,  a  class  having  no 
terms)  we  may  describe  the  variable  as  a  null  variable. 
For  example,  x  is  such  a  variable  if  it  denotes  any  integer 
greater  than  4  and  less  than  5. 

It  will  be  very  helpful  to  illustrate  the  notion  of  a 
variable  by  means  of  examples.  Before  doing  so,  how- 
ever, I  wish  to  handle  briefly  a  puzzling  question  that  may 
have  occurred  to  you  in  the  course  of  the  foregoing  discus- 
sion.    We  saw  that  the  admissible  terms  for   </>(#),— the 


240  MATHEMATICAL   PHILOSOPHY 

sense-giving  terms, — constitute  a  class  of  terms — the 
range  of  x.  Let  us  denote  the  class  by  C.  The  question 
is:  Do  the  nonsense-giving  terms, — the  inadmissible 
terms, — constitute  a  class?  The  answer  is  No;  there  are 
such  terms,  but  they  do  not  together  constitute  a  class. 
The  correctness  of  th$  answer  may  be  shown  as  follows: 
If  the  terms  in  question  constitute  a  class,  denote  it  by 
C;  C'y  being  itself  a  term,  is  either  in  C  or  in  C;  to  see 
that  C"  is  not  a  term  in  C,  consider  any  simple  example  of 
<f>(x) — say,  x  is  a  man;  in  this  case  C  is  composed  of  all 
the  terms  such  that  it  is  significant, — true  or  false, — to  say 
that  any  one  of  them  is  a  man;  our  hypothetical  C  con- 
sists of  all  the  terms  such  that  it  is  neither  true  nor  false, 
but  is  nonsense  to  say  that  any  one  of  them  is  a  man; 
evidently  it  is  neither  true  nor  false,  but  is  nonsense  to 
say,  "  The  class  of  all  the  terms  such  that  it  is  nonsense 
to  say  they  are  men  "  is  a  man;  hence  C,  if  there  be  such 
a  class,  is  not  a  term  in  C,  but  is  a  term  in  C;  it  is,  how- 
ever, foolish  to  say  that  C  is  a  term  in  C",  in  itself — as 
\  foolish  as  to  say,  for  example,  that  a  class  of  apples  or  of 
points  is  an  apple  or  a  point,  or  that  the  class  of  featherless 
1  /  bipeds  is  a  two-legged  thing  without  feathers.  And  thus 
you  see  that  the  inadmissible  terms  for  a  given  preposi- 
tional function  do  not  constitute  a  class.  The  question 
I  have  thus  summarily  treated  is  of  the  kind  of  questions 
which  have  led  Messrs.  Russell  and  Whitehead— or  rather 
have  driven  them — to  the  theory  of  Types  in  the  Principia. 
In  its  present  state  the  theory  is  far  from  being  entirely 
satisfactory,  but  it  is  exceedingly  helpful  and  it  undoubt- 
edly faces  in  the  right  direction.  I  desire  to  recommend 
it  to  you  for  consideration  and  for  improvement. 

Examples  of  Variables. — Let  us  now  turn  to  the  task  of 
illustrating  the   mathematical   notion   of  a   variable   by 


VARIABLES  AND  LIMITS  241 

means  of  various  examples.  Every  example  of  a  great 
idea  gives  a  little  light  and  casts  a  big  shadow;  we  must 
try  to  see  the  idea  in  the  mingled  lights  and  not  lose  it  in 
the  composite  dark  of  the  shadows.  Again  consider  the 
propositional  function  </>(#).  The  class  C — the  range  of 
the  function's  variable  x — is,  as  you  know,  the  logical 
sum  of  two  sub-classes:  Ci,  composed  of  the  verifiers  of 
<f>(x);  and  C2,  composed  of  the  falsifiers  of  <f>(x).  We 
may  chance  to  be  interested  only  in  the  true  propositions 
derivable  from  <f>(x)  or  only  in  the  false  ones;  accordingly 
we  then  restrict  our  thought  to  the  verifiers  or  else  to  the 
falsifiers;  if  to  the  former,  then  the  variable  x  represents, 
not  any  one  of  the  terms  in  C,  but  any  one  of  the  verifiers 
— x's  range  being,  not  C,  but  Ci;  if  to  the  latter,  then 
x's  range  is  C2.  In  these  cases  what  determines  the 
variable's  range?  The  answer  is:  neither  the  function 
alone  nor  our  restrictive  decision  alone,  but  the  two 
things  taken  together.  The  range  of  a  variable  is  in  every 
case  either  the  class  of  admissible  terms  for  some  propo- 
sitional function  or  a  sub-class  of  such  a  class,  the  sub- 
class being  determined  by  some  restriction  which  the 
function  as  such  does  not  impose;  observe  that,  if  some 
symbol  x  is  to  be  a  variable,  it  is  we  who  decide  what  its 
range  is  to  be,  for  it  is  we  who  choose  the  function  and, 
if  any  restriction  is  added,  it  is  we  who  impose  it.  Is  our 
freedom  in  the  matter  absolute?  No;  there  is  no  such 
thing  as  absolute  freedom;  in  the  matter  in  question,  as 
in  all  other  purely  intellectual  matters,  we  have  all  the 
freedom  there  is,  but  it  is  not  absolute;  we  have  just  seen 
that  we  can  not  have  a  variable  representing  "  any  one  " 
of  the  inadmissible  terms  for  a  given  propositional  func- 
tion, for  the  supposition  that  we  can  leads,  as  we  saw,  to 
contradiction.     Freedom   of   thought, — intellectual   free- 


242  MATHEMATICAL   PHILOSOPHY 

dom, — is  conditioned,  restricted,  limited;  but  it  is  funda- 
mentally limited  by  only  one  Law — the  law  which  says  to 
Intellect,  "  Thou  shalt  not  incur  a  contradiction  in 
terms."  This  law  is  the  eternal  guardian  of  intellectual 
integrity;  reverence  for  it, — the  disposition  to  keep  it, — 
is  the  absolute  invariant  of  intellectual  life;  disregard  of 
the  law, — I  do  not  mean  inadvertent  violation  of  it, — 
means  intellectual  extinction:  for  intellect,  disloyalty  is 
death.  Incidentally,  we  thus  glimpse  another  phase  of  the 
truth,  mentioned  before,  that  mathematics  is  the  study 
of  Fate  and  Freedom. 

Examples  illustrating  the  mathematical  concept  of  a 
variable  are  more  numerous  than  the  sands  of  the  seashore 
or  indeed  the  stars  of  the  heavens,  even  if  the  multitude 
of  the  stars  be  infinite.  In  examining  the  following  more 
specific  and  more  familiar  specimens  it  should  be  borne 
in  mind  that  in  mathematical  discourse  the  range  of  a 
variable  is  very  frequently  indicated  without  explicitly 
stating  the  propositional  function  or  functions  necessarily 
involved  in  determining  the  range;  such  statement  is, 
however,  always  possible  and  is  often  made.  And  now 
some  familiar  specimens. 

(i)  Consider  the  finite  cardinal  numbers:  0,  I,  2, 
3,  4,  5,  .  .  .;  let  x  represent  any  one  of  them;  here  the 
symbol  x  is  a  variable;  its  range  is,  not  the  endless  row 
as  such,  but  the  class  of  terms  (numbers)  in  the  row;  the 
range  is  the  same  as  would  be  indicated  if  we  said,  let  x 
represent  any  one  of  the  verifiers  of  the  propositional 
function — n  is  a  cardinal  number.  What  is  here  the 
range  of  n  ? 

(2)  A  variable's  range  may  be  finite  or  infinite.  In  (1) 
the  range,  you  note,  is  infinite.  If  we  let  x  represent  any 
cardinal  greater  than  1  and  less  than  10,  x's  range  is  the 


VARIABLES  AND  LIMITS  243, 

finite  class  composed  of  the  terms,  2,  3,  4,  5,  6,  7,  8,  9: 
we  can  indicate  the  range  by  saying,  let  x  represent  any 
one  of  the  verifiers  of  the  function — n  is  a  cardinal  whose 
value  is  between  1  and  10.  What  is  n's  range?  What 
would  the  range  of  x  be  if  we  simply  said  "  x  is  a  cardinal 
between  1  and  10  "  ? 

(3)  If  we  say,  let  x  be  any  verifier  of  the  function 
— n  is  a  cardinal  between  5  and  7 — ,  the  range  of  x  is — ,  not 
5 — but  the  class  whose  sole  member  is  5;  in  this  case  the 
varible  is  a  constant. 

(4)  Consider  the  infinite  (endless)  series:  l+2+3-f 
4  +  5+.  .  •  ;  the  sum,  S„,  of  the  first  n  terms  is  %n(n-\-i), 
so  that  Sn  =  %n(n  +  i).  Here  the  language  implies  that 
n  is  being  regarded  as  a  variable  whose  range  is  the  class 
of  all  positive  integers;  perhaps  some  one  doubts  the 
implication;  very  well,  let  us  explicitly  agree  to  let  n  be 
such  a  variable;  you  see  at  once  that  we  then  have 
another  variable  on  our  hands,  namely,  Sn — an  ordinary 
(not  a  propositional)  function  of  n,  for  to  each  value,  as 
we  say,  of  n  (i.e.,  to  each  number  in  n's  range)  there 
corresponds  a  definite  sum,  a  definite  value  of  S„;  plainly 
these  sums  are:  1,  3,  6,  10,  15,  .  .  .;  the  class  of  these  is 
the  range  of  the  variable  S».  Adopting  the  usual  figura- 
tive speech,  we  may  say  that,  as  the  variable  n  runs  along 
the  row  1,  2,  3,  .  .  .  ,  the  dependent  variable,  or  function 
S»y  glides  along  the  sequence  1,3,6,  .  .  .  But  we  must 
not  let  such  talk  make  us  forget  what  the  ranges  are: 
these  are  classes  and  not  rows  of  their  members  or  terms; 
the  terms  of  either  range  appear  in  many  different  rows, 
but  the  range  is  one  thing  and  each  of  the  rows  is  another. 
If  we  so  desire  we  can  make  explicit  the  propositional 
function  involved  in  determining  the  range  of  Sn;  we  can 
say,  for  example,  that  the  symbol  Sn  represents  any  one 


244  MATHEMATICAL    PHILOSOPHY 

of  the  terms  in  the  class  of  verifiers  of  the  propositional 
function — x  is  the  first  term  or  the  sum  of  the  first  two 
terms  or  the  sum  of  the  first  three  terms,  ...  of  the 
row  i,  2,  3,  4,  5,  ...  ;  such  speech  is  or  may  become  for- 
biddingly cumbrous,  but  it  serves  to  remind  us  that, 
though  propositional  functions  be  not  always  mentioned, 
they  are  yet  omnipresent  and  may,  if  we  so  desire,  be 
made  manifest  to  the  eye  or  the  ear. 

(5)  If  we  agree  to  let  x  represent  any  one  of  the 
numbers  in  the  unending  row,  I,  §,  \y  §,  .  .  .  ,  the  range 
of  the  variable  x  is  the  class  of  all  the  numbers  in  the  row; 
it  is  common  and  often  convenient  to  indicate  this  row 

by  writing:  1,  -,  —,—,—,...,  nm,  •  •  •  ;        the  symbol 

2   2     2     2*  2 

-^z"i  is  called  the  general  term  or  the  nth  term  (of  the  row); 

2 

it  is,  you  observe,  a  function  of  n.  But  what  is  n  here? 
Plainly  it  is  a  variable  whose  range  is  the  class  of  positive 

integers;    so  the  symbol   -^1  is  a  variable  depending  on 

another  variable  n;  using  our  figurative  way  of  speaking, 
we  may  say  that,  as  n  varies  from  term  to  term  of  its 

range,  the  function  ~^zi  runs  from  term  to  term  of  its 

2 

range,  the  class  of  numbers  in  the  foregoing  row;  we  may 
say  that,  if  n  starts  at  1  and  runs  along  the  row  1,2,..., 

—j—i  starts  at  1  and  runs  along  the  row  1,  |,  j,  .  .  .  ;    or 

we  may  say  more  precisely  that  n  represents  any  one  of 

the  numbers  of  its  range  and  that  -^1  correspondingly 

represents  any  one  of  the  numbers  of  its  range;  the  latter 


VARIABLES  AND  LIMITS  245 

range  is  identical  with  the  range  of  x,  and  so  you  see  that 
in  this  example  x  and  -^r  are,  as  variables,  identical,  two 

2 

symbols  playing  the  same  role. 

(6)  Consider  the  infinite   (unending)   series, 

III  T 

I+-  +  -+-+    .    .    . 


n  —  li 

2       2Z       2*  2 

its  terms  are  those  of  the  row  in  (5);  it  is,  you  notice,  a 
geometric  series  or  progression  whose  ratio,  as  it  is  called, — 
the  ratio  of  any  term  (except  the  first)  to  its  predecessor, — 
is  |;    let  us  denote  the  sum  of  the  first  n  terms  by  Sn; 

then,  as  you  learned  in  elementary  algebra,  Sn  =  2——^r\. 

2 

Observe  that  we  are  here  confronted  with  three  related 

variables:    ny  ~T-it  and  Sn,  the  second  being  a  function 

2 

of  the  first,  and  the  third  a  function  of  the  second  (directly) 
and  of  the  first  (indirectly,  through  the  second).  What 
are  their  respective  ranges?  That  of  the  first  is  the  class 
of  positive  integers;    that  of  the  second  is  the  class  of 

numbers  in  the  row  1,  -,—,—,...  ;   that  of  the  third  is 

2     2       2 

3   7   15 
the  class  of  numbers  in  the  row  1,  -,  -,  — ,  ...     I  leave 

2   4   16 

it  to  you  to  describe  the  situation  in  the  dynamic,   or 

picturesque,  language  of  change,  variation,  behavior. 

(7)  The  foregoing  six  examples  are  very  specific.     Let 

us   take   one   that   is    somewhat   less   so.     Consider   the 

geometric  progression 

(S)  a+ar2+ar3+  .  .  .  +arn~1+  .  .  .  ; 


246  MATHEMATICAL    PHILOSOPHY 

for  the  sum  of  the  first  n  terms  we  have,  as  you  know, 

I  r 


Sn=a 


I  —  r     i  —  r 


as  in  example  (6)  we  have  here  three  related  variables, 
n,  r"  and  SB,  if  we  suppose  (as  we  commonly  do)  that  a 
and  r  have  given  values;  and  you  may  wish  to  tell,  as  an 
exercise,  how  the  three  variables  are  related  and  what 
their  respective  ranges  are.  I  desire  to  call  your  attention 
to  the  fact  that,  when  we  say  "  Let  (S)  be  any  geometric 
progression,"  we  are  implicitly  treating  both  a  and  r  as 
variables,  and  not  only  that,  but  we  are  explicitly  treating 
(S)  itself  as  a  variable.  In  such  case,  what  are  the  ranges 
of  0,  r  and  (S)  ?  We  can  not  answer  definitely  until  we 
have  told  what  field  or  domain  of  number  we  are  working 
in,  as  (say)  that  of  the  positive  integers  or  that  of  the 
rational  numbers  or  that  of  all  the  so-called  real  numbers 
or  that  of  the  ordinary  complex  numbers,  and  so  on.  If 
we  are  working  in, — confining  ourselves  to, — the  domain 
of  the  real  numbers,  the  answer  is  that  the  range  of  a 
is  the  class  of  all  the  real  numbers,  that  r's  range  is  the 
same  as  as  and  that  the  range  of  the  variable  (S)  is  the 
class  of  all  unending  geometric  series  or  progressions 
formed  or  formable  of  real  numbers.  It  is  interesting  to 
observe,  in  passing,  that  the  range  of  (S)  contains  a  two- 
fold infinity  (°°2)  of  geometric  series;  for  assigning  some 
value  to  a,  we  can  obtain  as  many  series  whose  a  has  that 
particular  value  as  there  are  possible  values  for  r, — as 
many,  that  is,  as  there  are  real  numbers, — an  infinite 
multiplicity  often  denoted  by  the  symbol  °°  ;  it  is  plain 
that  for  each  of  these  we  get  <x>  series  by  letting  a  assume 
its  possible  values;  so  that  in  the  domain  of  real  numbers 
there  exist  oo  X  oo,  or  oo2,  unending  geometric  series — as 


VARIABLES  AND  LIMITS  247 

many  as  there  are  points  in  a  plane  or  pairs  of  real  num- 
bers. You  will  find  it  very  interesting  to  describe  in  your 
own  way  the  functional  relations  among  the  six  variables 
n,  a,  r,  rn,  Sn  and  (S),  and  very  instructive  to  state  explic- 
itly the  propositional  functions  implicitly  involved  in 
determining  the  various  ranges.  Instead  of  fixing  upon  a 
particular  domain  as  above,  you  may  say,  if  you  like,  "  Let 
our  number  domain  D  be  any  number  domain  ";  then  Dy 
too,  will  be  a  variable  and  the  complications  thicken. 
Variables,  you  see,  are  lurking  everywhere  and  are  ever 
ready  to  leap  into  the  arena  of  our  attention,  thus  lifting 
our  thought  to  higher  and  higher  levels  of  generality. 
The  scale  of  levels  is  summitless.  Science,  it  is  said,  is 
the  study  of  functions;  functions  are  variables — science 
is  the  study  of  variables;  every  variable  has  its  range 
partly  determined  by  the  scope  of  some  propositional 
function — science  is  the  study  of  propositional  functions; 
the  general  concept  and  the  name  of  propositional  function 
are  only  beginning  to  gain  a  little  recognition;  the  philoso- 
phy of  science  is  in  its  infancy;  the  infant's  nurse  is  the 
philosophy  of  mathematics;  in  the  richest  nation  known 
to  history, — made  such  by  science, — the  nurse  can  hardly 
contrive  to  live — she  can  hardly  even  publish  her  works 
because  they  are  not  profitable  commercially;  the  nation 
is  vain  and  boastful.     May  God  deliver  us. 

In  the  foregoing  list  of  examples  the  ranges  of  the 
variables  are  composed  of  numbers,  with  one  notable 
exception:  the  range  of  (S)  is  composed  of  series;  these 
are  indeed  constructed  out  of  numbers  but  they  are  not 
themselves  numbers;  and,  though  they  are  called  geomet- 
ric series,  they  are  not  spatial  entities  and  have  no  essen- 
tial connection  with  geometry.  In  citing  additional 
examples,  it  will  be  easy  to  include  among  them  some 


248  MATHEMATICAL   PHILOSOPHY 

that  are  strictly  geometric  and   some  that  are  neither 
geometric  nor  numerical. 

(8)  Consider  the  class  2  of  the  spheres  having  a  given 
point  C  for  center  and  any  radius  greater  than  zero. 
Each  of  the  spheres  has  many  points,  many  tangent  lines 
and  many  tangent  planes.  Each  sphere  has  an  interior, — 
the  region  or  room  bounded  by  the  sphere, — and  an 
exterior — the  region  outside  of  it.  If  you  let  S,  R,  R',  r, 
P,  Ly  ir,  I,  A,  V>  P',  P"  respectively  represent  any  one  of 
the  spheres,  any  one  of  the  interiors,  any  one  of  the 
exteriors,  any  one  of  the  radii  (taken  as  line  segments), 
any  one  of  the  points  on  any  one  of  the  spheres,  any  one 
of  the  tangent  lines,  any  one  of  the  tangent  planes,  any 
one  of  the  radial  lengths,  any  one  of  the  sphere  areas, 
any  one  of  the  sphere  volumes,  any  one  of  the  points 
common  to  all  the  interiors,  any  one  of  the  points  common 
to  all  the  exteriors,  then  the  symbols  S,  R,  R',  r,  P,  L,  iry 
P' ,  P"  are  geometric  variables,  and  /,  A,  V  are  arithmetic, 
or  numerical,  variables.  Of  the  former  ones  the  respective 
ranges  are:  the  class  S,  the  class  of  all  the  interiors  of  the 
spheres  in  2;  the  class  of  all  the  sphere  exteriors;  the 
class  of  all  the  radii;  the  class  of  all  the  points  of  our 
space  except  the  point  C;  the  class  of  all  the  lines  of  space 
except  the  lines  of  the  line-sheaf  having  C  for  vertex; 
the  class  of  all  the  planes  of  space  except  those  of  the 
bundled  vertexed  at  C;  the  class  whose  sole  member  is 
the  point  C  (Pf  being  thus  a  constant);  and  the  null 
(empty)  class  of  points  (the  range  of  P"  containing  no 
terms);  the  ranges  of  /,  A  and  V  are  the  same,  namely, 
the  class  of  all  the  positive  real  numbers  (zero  not 
included). 

Let  me  remind  you  of  something  and  then  submit  a 
few  questions  that  should  give  you  a  happy  hour  or  so  of 


VARIABLES  AND  LIMITS  249 

cloistral  meditation.  You  know,  I  believe,  that  when 
mathematicians  say  that  two  variables  V  and  V  (having 
K  and  K'  for  their  ranges)  are  functions  of  each  other, 
they  mean  that  to  any  given  term  in  either  range  there 
corresponds  a  definite  term  (or  terms)  in  the  other  range; 
you  know  that,  if  to  each  term  in  (say)  K'  there  corre- 
sponds only  one  term  in  K,  then  V  is  called  a  cm<?-valued 
function  of  V,  and  that,  if  two  or  more  terms  in  K  corre- 
spond to  a  term  in  K',  V  is  called  a  many-valued  function 
of  V ',  where  the  "  many  "  may  be  finite  or  infinite^ 
The  questions  I  wish  to  ask  are  these:  Is  each  one  of  the 
foregoing  12  variables,  S,  R,  R'>  r,  and  so  on,  a  function 
of  each  of  the  II  others?  Which  ones,  if  any,  of  these 
functions  are  one-valued?  Which,  if  any,  are  many- 
valued?  Which,  if  any,  are  infinitely  many-valued? 
What  sort  of  a  function  is  /  of  r?  r  of  /?  Is  there  in  2 
a  least  sphere?  A  largest  one?  If  we  choose  to  regard 
the  common  center  C  as  a  sphere  of  zero  radial  length, 
and,  in  describing  2,  drop  the  requirement  that  every 
radius  must  exceed  zero,  what,  if  any,  changes  must  be 
made  in  the  answers  to  the  foregoing  questions?  If  we 
fancy  ourselves  working  in  projective  space — which,  as  you 
know,  has  one  and  but  one  plane  (an  infinite  sphere)  at  00 
and  if  we  let  2  be  the  class  of  spheres  having  a  given 
common  center  C  and  radius  equal  to  or  greater  than 
zero,  what  are  the  ranges  of  our  12  variables? 


1  *  £    «    % 

Fig.  25. 

(9)  For  further  examples  of  variables,  some  of  them 
geometric,  some  of  them  numerical,  consider  the  following: 
Let  the  line-segment  AB  have  a  length  of  2  units.    Denote 


250  MATHEMATICAL   PHILOSOPHY 

by  Pi  the  mid-point  of  AB,  by  P2  the  mid-point  of  seg- 
ment PiB,  by  P3  the  mid-point  of  P2B,  and  so  on  end- 
lessly, so  that  Pn  will  denote  the  mid-point  of  the  segment 
Pn_1B.  Suppose  we  let  the  symbols  P,  S,  S',  I,  I'  represent 
respectively  any  one  of  the  indicated  mid-points,  any 
one  of  the  segments  {AP\,  P1P2,  P2P3,  •  .  .  ),  any  one 
of  the  segments  beginning  at  A  and  ending  at  a  mid- 
point, any  one  of  the  lengths  of  the  S-segments,  any  one 
of  the  lengths  of  the  S'-segments;  the  ranges  are  respect- 
ively the  class  of  the  mid-points,  the  class  of  segments  in 
the  row  A  Pi,  P1P2,  P2P3,  ■  ■  •  ,  the  class  of  segments  in  the 
row  APu  AP2,  AP3,  AP±,  .  .  .  ,  the  class  of  numbers  in 

tl»e  row  I,  — ;  -r-,  — ,  .  .  .  ,  and  the  class  of  numbers  in  the 

2    2    2 

row  1,  —,  '—  .  .  .    Observe  that  P,  S  and  S"  are  geometric 

2  4 

variables  and  that  /  and  /'  are  numerical.  Observe  that 
the  role  of  P  is  the  same  as  that  of  Pn  and  so  we  may  write 
P  =  Pn;  similarly,  we  may  write  S=Pa_1Pn  (understanding 

that  Po  is  A),  S'  =  APn,  l  =  ~T=ri  and  l'  =  2 5^1;  and  so, 

2  2 

you  see,  the  variables  P,  S,  S',  /,  /'  are  functions  of  the 
variable  n  whose  range  is  the  class  of  positive  integers. 
Are  the  functions  one-valued  or  are  they  many-valued? 
Describe  the  so-called  variation  of  these  functions  as  n 
"  varies."  As  n  increases  more  and  more  will  Pn  ever 
reach  P?  Why  not?  Can  you,  by  increasing  n,  make 
the  length  of  the  segment  PnB  smaller  than  any  length 
you  choose  if  you  don't  choose  zero  length  ? 

(10)  In  Fig.  26  we  have  a  circle  K  of  given  center  C 
and  radius  of  given  length  R.  By  circle  K,  I  mean  the 
curve.  Denote  the  length  of  K  by  L  and  the  area  of  K  by 
A.     Do  not  fail  to  distinguish  geometric  from  numerical 


VARIABLES  AND  LIMITS 


251 


things:  observe,  for  example,  that  K  is  geometric, — a 
specific  curve, — and  that,  like  R,  L  and  A  are  numerical — 
specific  numbers.  Consider  the  indicated  inscribed  regu- 
lar polygons — the  triangle  (or  3-side),  hexagon  (or  6-side), 
do-decagon  (or  12-side),  24-side,  48-side,  and  so  on  end- 
lessly; note  that  the  nx\\  polygon  of  the  unending  row  has 


Fig.  26. 


n-l 


3X2""  sides.  You  see  that  n  is  a  variable  whose  range 
is  the  class  of  positive  integers.  Denote  by  Pu  the  nth 
polygon,  by  Lu  the  perimeter  (length)  of  P„,  by  An  the 
area  of  (size  or  magnitude  of)  Pn,  and  by  dn  the  length  of 
the  perpendicular  from  C  to  a  side  of  Pn.  The  symbols 
Pn,  Ln,  An  and  da  are  variables.  Which  ones  are  geo- 
metric?    Which   numerical?     They    are   functions   of  n. 


252  MATHEMATICAL   PHILOSOPHY 

What  kind?  What  are  their  ranges?  Are  these  finite 
or  infinite?  Do  the  functions  increase  or  decrease  with  w? 
All  increase  with  n  except  P»;  how  justify  the  exception? 
Can  n  so  increase  that  Pn  will  coincide  with  K\  Or  Ln 
with  Z,?  Or  An  with  At  Or  dn  with  R?  Why  not? 
the  differences  L—Ln,  A  —  A„,  R—dn  are  variables,  func- 
tions of  n — are  they  not?  If  I  name  a  positive  number  a 
as  small  as  I  please,  can  you  then  choose  such  a  value  for 
n, — such  a  term  in  ns  range, — that  the  foregoing  differ- 
ences will  be  less  than  <r?  Having  thus  chosen  n,  if  you 
let  n  take  still  larger  and  larger  values,  will  the  differences 
in  question  keep  always  less  than  <r?  Will  they  ever 
vanish — that  is,  be  equal  to  zero?     Why  or  why  not? 

(n)  Let  us  now  consider  two  variables  that  are 
neither  geometric  nor  numerical.  Referring  to  Fig.  26, 
let  us  write  K  —  P\,  K  —  P2,  .  .  .  ;  these  seem  to  be  symbols 
but  what,  pray,  do  they  symbolize?  They  are  indeed 
suggestive,  but  they  have  at  present  no  definite  meanings, 
for  neither  K  nor  any  of  the  P's  is  a  number,  K  being  just 
a  circle — a  specific  circle — and  each  P  a  specific  polygon. 
We  may,  however,  assign  meanings  to  the  ostensible 
symbols,  thus  making  them  genuine  symbols.  What 
meanings  shall  we  assign?  A  little  reflection  upon  the 
make-up  of  Fig.  26  naturally  suggests  that  we  let  K  —  Pi, 
K  —  P2,  K  —  Pz,  .  .  .  ,  denote  respectively  the  endless 
polygon-row.?  indicated  at  the  right  of  them  below: 


(S) 


K-P, 
K-P2 
K-P3 


P2,  Pz,  P4,  P5, 
P3,  P±,  P5,  .  . 
P±,  P5, 


VARIABLES  AND  LIMITS  253 

Note  that  we  have  here  an  endless  (downward  running) 
row  (S),  of  endless  rows  of  polygons.  It  is  obviously  natu- 
ral to  denote  by  K  —  Pn  any  one — the  nth  one — of  the 
rows  in  the  row  (S).  The  symbol  K — Pn  thus  becomes 
a  variable  whose  range  is,  not  (S),  but  the  class  of  rows 
in  (S).  Our  new  variable  is  plainly  not  numerical; 
neither  is  it  geometric  for,  though  the  rows,— the  terms  in 
its  range, — are  composed  of  polygons,  a  row  of  geometric 
figures  is,  in  strictness,  no  more  a  geometric  figure  than  a 
row  of  men  is  a  man. 

For  an  example  of  a  variable  more  evidently,  though 
not  more  actually,  non-geometric  and  non-numerical  than 
the  preceding  one,  we  may  take  p  where  p  represents  any 
one  of  the  propositions  derivable  from  some  given  prepo- 
sitional function  </>(#)  by  means  of  its  verifiers.  The 
range  of  p  is  obviously  the  class  of  true  propositions 
having  4>(x)  for  matrix. 

Conceptions  of  Limit. — It  is  evident  that  much  of  our 
human  thinkings — I  strongly  suspect  that  all  of  it, — is 
concerned  with  variables.  One  of  the  lessons  which  the 
history  of  thought  and  our  personal  experience  in  thinking 
teach  very  clearly  is  that  we  can  not  deal  with  variables 
logically  or  with  any  close  approximation  to  rigor  without 
the  help  of  the  notion  (or  notions)  which  mathematicians 
denote  by  the  term  "  limit."  We  must,  accordingly,  try 
to  understand  what  the  term  means.  I  have  just  now 
used  the  plural— conceptions  of  limit — instead  of  the 
singular.  I  have  done  so  because  there  are  various 
meanings  of  the  term  and  I  intend  to  present  more  than 
one  of  them.  The  definitions  I  am  going  to  present  are 
closely  related,  but  they  are  not  equivalent:  they  differ 
in  content  and  scope,  and  you  will  find  it  very  instructive 
to  compare  them  in  these  respects.     You  will  observe  at 


254  MATHEMATICAL   PHILOSOPHY 

once  that  there  are  two  respects  in  which  they  agree — 
they  involve  the  notion  of  variable  and  its  range  (every 
limit  is  a  limit  of  a  variable)  and  all  but  one  of  them 
involve  the  notion  of  difference.  The  notion  of  variable 
we  have  discussed  at  unusual  length  and  it  is,  I  trust,  now 
fairly  clear.  It  will  be  useful,  I  believe,  to  say  a  pre- 
liminary word  regarding  the  notion  of  difference.  It  is 
not  my  aim  to  define  the  general  notion;  my  aim  is  merely 
to  enliven  a  little  our  consciousness  of  it.  In  the  back- 
ground of  our  human  thinking,  however  refined,  however 
precise  the  ideas  we  are  explicitly  handling,  there  lurk 
other  ideas — shadowy,  nebulous,  vague — which  we  have 
not  defined,  which  we  may  not  have  attempted  to  define, 
which  we  may  not  even  be  conscious  of;  yet  these  back- 
ground ideas  give  our  so-called  precise  ones  all  the  meaning 
the  latter  have.  In  the  present  discussion,  the  idea  of 
difference  is  a  background  idea. 

Any  given  variable,  as  we  have  seen,  has  a  range — a 
certain  class  of  things,  or  objects,  called  the  terms  of  the 
class  and  commonly  spoken  of  as  the  variable's  "  values." 
A  class  being  given,  each  of  its  terms  is  comparable  with 
each  in  one  or  more  respects:  that  is  to  say,  each  of  them 
differs  from  each  in  one  or  more  respects;  the  respects, 
and  hence  the  differences,  may  be  very  definite  or  fairly 
definite  or  very  vague;  the  differences  may  be  differences 
in  respect  of  position  or  of  magnitude  (size)  or  of  number 
or  color  or  shape  or  weight  or  of  importance  or  of  dignity 
or  of  beauty  or  of  sensibility  and  so  on;  we  may,  therefore, 
speak  of  kinds  of  difference  as  distinguished  from  amounts 
of  a  given  kind.  It  is  essential  to  note  the  obvious  fact 
that,  if  each  term  of  a  class  differs  from  each  of  its  fellow 
terms  in  respect  to  some  specific  kind  k  of  difference,  it 
may  happen  that  the  terms  in  the  class  differ  from  some 


VARIABLES  AND  LIMITS  255 

terms  not  in  it  in  respect  to  the  same  kind  k.  Examples 
abound,  and  need  not  be  cited.  If,  with  respect  to  a  given 
kind  of  difference,  two  terms  be  identical,  we  shall  say 
that  the  amount  of  their  difference  (of  the  given  kind) 
is  null  or  naught;  if,  as  often  happens,  the  given  kind  of 
difference  be  numerical,  then  the  hypothetical  identity 
is  numerical  equality  and  we  shall  say,  in  accordance  with 
usage,  that  the  amount  of  difference  is  zero;  we  shall  thus 
be  using  zero  as  a  special  variety  of  the  foregoing  null  or 
naught.  Here,  as  elsewhere,  it  is  essential  to  use  good 
sense.  It  would,  for  example,  be  nonsense  to  speak  of 
two  numbers  as  differing  in  respect  to  color  or  patriotism 
or  loyalty.  When  we  speak  of  two  terms  as  having  an 
amount  of  difference  of  a  kind  k,  it  is  to  be  understood  that 
k  is  a  kind  with  respect  to  which  the  terms  can  be  sig- 
nificantly compared.  A  kind  k  of  difference  may  be 
called  a  ^-difference;  and  a  given  amount  d  of  it,  the 
^-difference  d. 

I  am  now  going  to  define  a  very  convenient  idea  to  be 
called  a  k-neighborhood  of  a  term  t.  Let  t  be  a  term 
comparable  with  one  or  more  terms  in  respect  to  a  kind  k 
of  difference,  and  let  d  be  a  given  amount  (greater  than 
null)  of  such  difference;  then  d  is  said  to  determine  a 
^-neighborhood  of  t.  If  d  grow  larger  or  smaller,  the 
neighborhood  will  do  likewise.  If  d  be  specified,  we  may 
speak  of  the  ^-neighborhood  d  of  t.  Observe  that  d 
determines  the  same  neighborhood  for  each  and  every  term 
comparable  with  one  or  more  terms  in  respect  to  the 
given  difference-kind  k.  A  term  t'  will  be  said  to  be  in 
or  not  in  the  ^-neighborhood  d  of  t  according  as  t'  differs 
from  t  by  an  amount  less  than  or  not  less  than  d.  You 
see  at  once  that,  if  t'  be  in  the  ^-neighborhood  d  of  /,  t  is 
in  the  ^-neighborhood  d  of  /'.     Obviously  t  is  itself  in  all 


256  MATHEMATICAL   PHILOSOPHY 

the  ^-neighborhoods  obtainable  by  letting  d  vary,  for  the 
amount  of  ^'s  difference  from  t  is  null  and  null  is  less  than 
any  amount  d  is  allowed  to  be. 

Let  us  now  give  the  following  broad  definition  D\  of 
the  term  limit,  and  see  what  happens. 

D\ :  Let  V  be  a  variable,  R  its  range,  and  k  a  kind  of 
difference  with  respect  to  which  the  terms  in  R  are  com- 
parable; if  there  be  a  term  /  (in  R  or  not)  such  that,  how- 
ever small  a  ^-neighborhood  of  t  be  chosen,  some  term  of 
R  is  in  the  neighborhood,  then  t  is  a  &-limit  of  V. 

When,  as  is  usual,  the  context  indicates  what  particular 
k  is  under  consideration,  it  need  not  be  mentioned  explic- 
itly and  we  may  speak  of  a  t  (satisfying  the  foregoing 
conditions)  simply  as  a  limit  of  V. 

It  is  easy  to  see  that,  under  definition  Di,  every  term 
in  the  range  R  of  a  variable  V  is  a  limit  of  V,  for,  if  t  be 
in  R,  and  k  be  an  admissible  kind  of  difference,  the  amount 
of /'s  /e-difference  from  t  being  null,  t  is  in  every  ^-neighbor- 
hood of/,  and  is,  therefore,  a  £-limit  of  V.  A  null  variable 
has  no  limit;  every  other  one  has.  Unless  the  contrary 
be  clearly  indicated,  let  us  understand  that  the  variables 
under  discussion  are  not  null  variables.  Can  a  V  have  a 
limit  that  is  not  in  the  R  of  VI  If  t  be  not  in  R  and  if 
fs  ^-difference  from  some  term  in  R  be  null,  then  clearly 
t  is  a  &-limit  of  V;  if  an  outside  t  be  a  &-limit  of  V>  it 
may  or  may  not  be  a  £Mimit  of  V,  if  k  and  k'  be  not  the 
same.  For  example,  let  V's  range  have  but  one  term — 
say,  a  sphere  S  of  given  color,  mass  and  volume;  let 
^-difference,  ^'-difference  and  ^"-difference  be  respectively 
difference  in  color,  in  mass  and  in  volume;  then  any 
object  having  the  same  color  as  S  will  be  a  &-limit  of  V, 
though  not  in  general  a  £'-limit  or  a  F'-limit;    and  so  on. 

Query:    Can  a  V  have  a  &-limit  t  not  in  the  R  of  V 


VARIABLES  AND  LIMITS  257 

if  the  amounts  of  t's  ^-difference  from  the  terms  in  R  be 
each  more  than  null?  If  R  be  a  finite  class,  the  answer 
is  evidently  no.  If  R  be  an  infinite  class,  the  answer 
depends  on  k.  Here  we  must  make  an  important  distinc- 
tion. The  difference-kind  k  may  be  such  that,  given  any 
two  amounts  of  it,  there  are  at  most  only  a  finite  number 
of  intermediate  amounts.  Such  a  kind  may  be  called  a 
discrete  difference-kind.  An  example  is  the  difference- 
kind  we  have  or  should  have  in  mind  when,  confining  our 
attention  to  zero  and  the  positive  integers,  we  talk  of  the 
"  differences  "  (strictly,  the  amounts  of  difference)  found 
by  subtraction;  of  such  amounts,  the  smallest  is  zero, 
the  next  smallest  is  I,  the  next  2,  and  so  on,  and  there  are 
no  other  amounts  of  the  kind  of  difference  we  are  here 
dealing  with.  If  we  were  talking  of  "  differences," — 
amounts  of  difference — of  (say)  rational  fractions,  we 
should  have  in  mind  a  different  kind  of  difference.  As 
in  the  foregoing  example,  so  if  k  be  any  discrete  difference- 
kind,  there  is,  as  you  readily  see,  an  amount  of  the 
^-difference  next  greater  than  the  null  of  it.  On  the  other 
hand,  a  difference-kind  may  be  such  that,  given  any  two 
amounts  of  it,  there  is  one  amount  (and  hence  infinitely 
many  amounts)  intermediate  to  the  given  ones.  Such  a 
kind  may  be  called  a  compact  or  dense  difference-kind.  An 
example  is  the  difference-kind  involved  when,  confining 
our  attention  to  the  rational  numbers,  we  say  the  amount 
of  difference  of  this  fraction  and  that  is  so-and-so.  It  is 
perfectly  clear  that  no  amount  of  dense  difference-kind  is 
next  greater  than  the  null  amount  of  it. 

Let  us  now  return  to  our  query.  If  the  ^-difference 
be  discrete,  the  answer  is  negative,  even  though  R  be 
infinite.  For  let  d  be  the  smallest  amount  of  ^-difference 
except  null;  then  no  term  t'  of  R  is  in  the  /('-neighborhood 


258  MATHEMATICAL   PHILOSOPHY 

d  of  t  for,  if  *'  were  in  the  neighborhood,  then  the  amount 
of  ^-difference  of  t  and  t'  would  be  less  than  d,  but  there  is 
no  such  amount  except  null,  and  null  is  excluded  by  the 
hypothesis  of  the  query.  On  the  other  hand,  if  the 
^-difference  be  dense,  the  answer  is  affirmative:  there  are 
F's  having  the  sort  of  limit  required.  For  it  is  sufficient 
that  the  range  R  be  such  that  for  some  t  not  in  R  there  is, 
for  any  chosen  ^-neighborhood  d  of  t,  however  small  the 
neighborhood,  an  R  term  in  the  neighborhood — that  is, 
an  R  term  differing  from  t  by  more  than  null  and  less  than 
d;  and  the  existence  of  such  F's  may  be  shown  by  letting 
V  be  the  variable  whose  R  has  for  its  terms  zero  and  all 
ordinary  fractions  less  than  I ;  you  note  that  I  is  a  t  not  in 
R;  that  in  comparing  the  terms  in  question  we  employ 
a  dense  difference-kind;  that,  however  small  a  neighbor- 
hood of  I  be  chosen,  R  has  terms  in  the  neighborhood; 
and  that  I  is,  therefore,  a  limit  of  V.  For  another  example 
consider  the  variable  x  in  (5)  of  our  foregoing  list  of 
variables;  show  that  zero,  which  is  not  in  the  range,  is  a 
limit  of  x.  You  will  find  it  interesting  and  very  instruct- 
ive to  examine  all  the  variables  of  the  cited  list  with  a 
view  to  ascertaining  which  of  them  have  limits  outside 
their  ranges  and  what  the  limits  are. 

The  concept  of  a  limit  as  defined  by  the  definition  D\ 
has,  you  see,  some  striking  properties.  It  is,  however, 
too  broad  for  certain  highly  useful  purposes;  for  example, 
it  does  not  sufficiently  discriminate  variables  among 
themselves;  according  to  it  all  variables  (except  null  ones) 
have  limits,  as  we  have  seen,  and  every  term  in  a  variable's 
range  is  a  limit  of  the  variable.  It  is  obviously  desirable 
to  classify  variables  with  respect  to  the  character  or  con- 
stitution of  their  ranges.  Let  us  accordingly  try  a  some- 
what narrower  definition  D2  of  the  term  limit. 


VARIABLES  AND  LIMITS  259 

D2I  Let  V  be  a  variable,  R  its  range,  and  k  a  suitable 
kind  of  difference;  if  there  be  a  term  t  (in  R  or  not)  such 
that,  however  small  a  ^-neighborhood  of  t  be  chosen,  some 
R  term  differing  from  2  by  more  than  null  is  in  the  neigh- 
borhood, then  t  is  a  £-limit  of  V. 

Note  that  the  sole  distinction  between  D\  and  A>  is 
due  to  the  presence  in  D2  of  the  phrase — "  differing  from 
/  by  more  than  null."  The  distinction,  seemingly  slight, 
is  very  grave,  as  we  shall  see.  In  the  first  place  we  easily 
see  that  no  V  has  a  £-limit  if  k  be  a  discrete  difference- 
kind.  For  suppose  t  to  be  such  a  limit;  let  d  be  the 
smallest  amount  of  the  /^-difference  greater  than  the  null 
amount;  then  R  has  a  term,  say  t',  in  the  ^-neighborhood 
d  of  t;  and  t'  differs  from  t  by  more  than  null  and  less  than 
dy  which  is  impossible.  Again,  no  V  whose  R  is  finite 
has  a  limit.  For  suppose  such  a  V  to  have  a  limit,  say 
t\  let  the  difference-kind  k  be  dense — the  preceding 
proposition  makes  it  superfluous  to  consider  the  case  of 
k  discrete;  the  number  of  terms  in  R  is  finite,  say,  n\ 
there  are  at  most  n  non-null  amounts  of  ^-difference 
between  them  and  t;  one  of  these  amounts,  say  d,  is  as 
small  as  any  of  them;  by  supposition  there  is  an  R  term, 
say,  t',  in  the  ^-neighborhood  d  of  t,  and  t'  differs  from 
t  by  more  than  null  and  less  than  d,  which  is  impossible. 

From  the  two  propositions  just  now  proved  it  follows 
that  if  a  V  have  a  limit  under  definition  D2,  it  is  necessary 
both  that  Vs  range  be  infinite  and  that  the  difference- 
kind  concerned  be  dense.  Are  these  necessary  require- 
ments also  sufficient?  A  simple  example  will  suffice  to 
show  that  they  are  not.  Let  F's  R  be  the  class  whose 
terms  are  the  rational  numbers:  0,  1,  2,  3,  4,  .  .  .  ;  the 
appropriate  difference-kind  is  dense  (since  we  are  dealing 
with  rationals)  and  R  is  infinite;  if  V  have  a  limit  /,  Ms  a 


260  MATHEMATICAL   PHILOSOPHY 

rational  number;  of  all  the  non-null  (non-zero  in  this 
example)  amounts  of  difference  between  t  and  the  R 
terms,  one  of  them,  say  d,  is  as  small  as  any  of  them, 
obviously;  hence  no  R  term  differing  from  /  by  more  than 
zero  is  in  the  neighborhood  d  of  t;  and  so  V  is  limitless. 

Query:  Are  there  any  variables  which,  under  Do,  have 
limits?  The  answer  is  yes;  such  variables  abound  in 
endless  number  and  variety.  Is  the  x  in  (i)  of  our  list  of 
variables  one  of  them?  No;  for  the  difference-kind 
appropriate  to  cardinal  numbers  is  discrete.  For  a  like 
reason   neither  is  the  variable   Sn  in   (4)   one  of  them. 

But,   as  you  easily  see,  the  variable  x,  or  -^zri,  in  (5)  has 

a  limit,  namely,  zero  (0);  so  has  Sn  in  (6),  the  limit  being 
2;  so,  too,  has  Sn  in  (7)  if  r  be  numerically  less  than  1; 
for  the  range  of  n  is  the  class:    1,  2,  3,  .  .  .  ;   the  range  of 

rn     . 
rn  is  the  class:    r1,  r2,  r3,  .  .  .  ;    the  range  of  -  —   is  the 

2  3 

class : , , , .  .  .  ;  and  the  range  of  Sn  is  the  class 

I  —  r   1  —  r   1  —  r 


a 


/    1  r 


\\  —  r     1  —  r)       \i  —  r     1  —r)       \i  —  r     I  —  r 


Consider  these  ranges.  Let  d  be  any  given  number 
greater  than  0;  as  r  is  less  than  1,  it  is  plain  that  there  are 
terms  in  the  range  of  r  exceeding  0  by  less  than  d — terms 
that  is,  in  any  prescribed  neighborhood  of  0,  however 
small;   and  so,  0  is  the  limit  of  r";   evidently  it  is  also  the 

rn 
limit  of  -— - ;    without  further  talk  you  see  that  in  any 

prescribed  neighborhood  of   — —  there  are  terms  of  the 

range  of  Sn;    hence  Sn  has  — —  as  limit.     What  if  r  be 


VARIABLES  AND  LIMITS  261 

not  numerically  less  than  I?  If  r  =  l,  the  divisor  i—r 
is  zero,  but  the  phrase  "  division  by  zero  "  is  meaningless, 
and  so,  if  r  =  I,  <S»  is  limitless.  If  r  be  numerically  greater 
than  i,  you  readily  see  that  the  (numerically)  smallest 

difference  between  — —   and  the  terms  of  S»'s  range  is 


■r 


z~~~\    hence,   if  we   choose   a   neighborhood   d  of    - 


CLT 

where  d  is  smaller  than  (and  we  can  take  it  thus 

i  —  r  v 

smaller,  if,  as  we  are  assuming,  a  is  not  zero),  SVs  range 

has  no  term  in  the  neighborhood;   therefore  — —  is  not  a 

limit  of  S„,  if  r  be  numerically  greater  than  zero.  Has 
Sn  any  limit  at  all  under  the  hypothesis?  You  can 
readily  show  that  it  has  not. 

I  wonder  if  you  are  growing  weary  of  this  long  discus- 
sion. I  must  believe  you  are,  unless  your  desire  to  under- 
stand the  subject  be  genuine,  deep,  invincible.  If  we  fail 
to  master  some  important  idea,  what  is  the  explanation? 
It  may  be  stupidity;  it  is,  more  probably,  unwillingness 
to  pay  the  price — meditation;  but,  if  we  be  really  inter- 
ested, meditation  is  not  a  price,  it  is  a  pleasure — a  sus- 
taining joy;  the  great  source  of  success  is  abiding  interest. 
I  have  been  counting  upon  your  interest  and  am  going 
to  count  upon  it  yet  further. 

Consider  that  somewhat  unfamiliar  variable  (S)  in 
(7)  of  our  list.  Has  (S)  a  limit?  Its  range,  we  saw,  is 
the  twofold  infinitude  of  geometric  series  formable  from 
a  and  r  where  a  and  r  are  variables  each  representing  any 
real  number.  As  the  range  of  a  (or  of  r)  is  the  class  of 
real  numbers,  it  is  evident  that  every  real  number  is  a 
limit  of  a  (and  of  r),  for  it  is  evident  that  in  any  given 


262  MATHEMATICAL    PHILOSOPHY 

neighborhood,  however  small,  of  any  given  real  number, 
there  are  real  numbers  whose  numerical  difference  from 
the  given  number  is  more  than  zero.  If  we  are  to  talk  of 
(S)  as  having  or  not  having  a  limit,  we  must  indicate 
what  we  are  going  to  mean  by  "  difference  "  of  geometric 
series  as  such.  This  we  may  do  as  follows:  let  {a\,  r\) 
be  a  given  pair  of  real  numbers,  and  let  d  and  d'  be  given 
positive  numbers;  we  agree  to  say  that  the  neighborhood 
d  of  a i,  and  the  neighborhood  d'  of  n,  together  determine  a 
neighborhood  (d,  d')  of  the  series  a\-\-a\r\-\-a\r\2  +  .  .  , 
and  that,  if  a-i  be  in  the  first  neighborhood  and  r2  in  the  sec- 
ond, then  and  only  then  the  series  a2+a2^2+«2^22+  .  .  .  , 
is  in  the  third.  Now,  as  we  saw  a  moment  ago,  a\  is  a 
limit  of  <z,  and  r\  of  r;  hence,  the  series  a\-\-a\T\-\- 
a\T\2-\-  ...  is  a  limit  of  a-\-ar-\-ar2-\-  .  .  .  ,  that  is,  of 
(S);  hence,  every  series  in  the  range  of  (S)  is  a  limit  of  (S). 
You  will  recall  that  in  (n)  of  our  little  list  of  variables, 
p  is  a  variable  whose  range  is  the  class  of  all  the  true 
propositions  having  a  given  propositional  function  4>{x) 
for  their  common  matrix.  Can  we  associate  the  notion 
of  limit  with  p\  We  can,  as  follows:  Let  V  be  a  variable 
whose  range  R  is  the  class  of  verifiers  of  (x);  denote  the 
range  of  p  by  R';  if  x\  be  a  given  term  in  R,  then  the 
proposition  4>(xi)  is  a  definite  term  in  R' — to  each  R 
term  there  thus  corresponds  an  R'  term,  and  conversely; 
let  k  be  a  suitable  difference-kind  for  the  R  terms;  it  will 
evidently  be  a  suitable  difference-kind  for  the  R'  terms, 
for,  if  and  only  if  the  amount  of  ^-difference  between  the 
R  terms  x\  and  xi  be  null,  the  corresponding  R'  terms 
4>(xi),  0(^2)  are  identical  propositions — indistinguishable 
with  reference  to  k;  we  will  regard  the  amount  of  /^-dif- 
ferences between  x\  and  X2  as  the  measure  of  ^-difference 
between  4>(*i)  an^   0(^2);    let  t  be  a  term  (in  R  or  not) 


VARIABLES  AND  LIMITS  263 

comparable  with  the  R  terms  with  respect  to  k;  then 
<£(X),  whether  in  R'  or  not,  is  comparable  with  the  R' 
terms  with  respect  to  k;  to  the  ^-neighborhood  d  of  t 
corresponds  the  ^-neighborhood  d  of  <i>(t);  if  an  R  term 
t'  be  in  the  former  neighborhood,  the  corresponding  R! 
term  4>{t')  is  in  the  latter  neighborhood;  and  so,  you  see, 
if  t  be  a  £-limit  of  V,  the  proposition  4>(t)  is  a  £-limit  of  p. 
Observe  that,  according  as  a  £-limit  of  V  is  or  is  not  an 
R  term,  the  corresponding  limit  of  p  is  a  true  or  a  false 
proposition. 

For  a  very  simple  example  of  the  foregoing,  suppose 
<t>{x)  to  be:  x  is  a  term  in  the  class  a  of  the  rational  num- 
bers |,  |,  j,  .  .  .  ;  ^'s  range  is  a;  ^>'s  range  is  the  class  a' 
of  propositions:  \  is  a  term  in  «;  |  is  a  term  in  «;  j  is  a 
term  in  a;  ...  .  Zero  being  a  limit  of  j^,  the  false  propo- 
sition— zero  is  a  term  in  a — is  the  corresponding  limit  of  p. 
What  change  of  supposition  will  make  the  limit  of  p  a 
true  proposition? 

Permit  me  to  recommend  strongly  that,  as  an  exercise, 
you  determine  which  of  the  variables  in  the  above-given 
list  of  variables  have  limits  under  definition  D2  and  what 
the  limits  are;  that  you  similarly  examine  a  goodly  variety 
of  variables  not  in  the  list;  and  that  you  consider  the 
question:  if  a  V  has  a  £-limit  t  under  D2,  has  it  the  same 
limit  under  Z)i? 

There  are  two  reasons  why  I  am  inviting  you  to  con- 
sider various  non-equivalent  definitions  of  the  term  limit. 
One  of  the  reasons  is  that  such  consideration  helps  to 
deepen,  refine  and  clarify  our  understanding  of  the  great 
conceptions — variable  and  range  thereof.  The  other 
reason  is  that  mathematicians  use  the  term  "  limit  "  in  a 
variety  of  senses  differing  in  scope.  In  any  discussion 
involving  the  term  "  limit,"  mathematicians,  when  they 


*64  MATHEMATICAL   PHILOSOPHY 

speak  carefully  (which,  being  human  beings,  they  do  not 
always  do),  indicate  explicitly  or  contextually  the  sense  in 
which  the  term  is  being  employed.  I  have  now  pre- 
sented two  widely  differing  definitions  of  the  term — 
D\  and  D2.  I  am  not  aware  that  the  former  one  has  been 
hitherto  given.  On  the  other  hand,  D2  (or  some  virtual 
equivalent  otherwise  stated)  is  of  very  frequent  use  in  the 
mathematical  literature  of  the  last  half-century.  In  the 
next  lecture  I  shall  invite  you  to  consider  additional  mean- 
ings of  the  term  in  question. 


LECTURE  XIV 
More  About  Limits 

FURTHER  DEFINITIONS  OF  LIMIT LIMITS   AND  THE  INFINI- 
TESIMAL CALCULUS CONNECTION  WITH  ORDER,  SERIES 

AND  SEQUENCES LIMITS  AND  LIMIT  PROCESSES  OMNI- 
PRESENT AS  IDEALS  AND  IDEALIZATION  IN  ALL  THOUGHT 
AND  HUMAN  ASPIRATION — IDEALS  THE  FLINT  OF 
REALITY GENIUS   AND   GENERALIZATION. 

There  are  two  additional  definitions  of  the  term 
"  limit  "  with  which  it  is,  I  believe,  very  important  for 
philosophical  students  to  get  well  acquainted.  Both  of 
them  are  closely,  indeed  essentially,  connected  with  what 
mathematicians  variously  call  a  linear  order  or  a  serial ' 
relation  or  a  series  or  a  sequence.  Before  presenting  them 
we  must  recall  clearly  to  mind  some  matters  briefly 
explained  in  Lecture  X  and  then  join  therewith  certain 
kindred  ideas  and  distinctions.  You  will  recall  that  a 
propositional  function,  say  </>(#,  y),  containing  two 
variables,  is  said  to  determine  a  (dyadic)  relation;  that, 
if  <t>(xi,  j2)  is  a  true  proposition,  then  and  only  then  the 
pair  or  couple  (#1,  ^2)  is  called  a  constituent  or  element  of 
the  relation;  that  the  class  of  all  such  constituents, — the 
class  of  all  the  pairs  verifying  (satisfying)  </>(*,  y),  is  the 
relation;  that,  if  we  denote  the  relation  by  Ry  we  say 
"  x  has  the  relation  R  to  y  "  by  writing  xRy;  that  R 
accordingly  has  a  sense — so  that,  if  (xi,  yi)   be   a  con- 

265 


266  MATHEMATICAL   PHILOSOPHY 

stituent,  we  have  xiRyi,  but,  in  general,  not  yiRxi;  that 
the  domain  of  R  is  the  class  of  all  the  x's, — the  class  of  all 
the  terms, — such  that  each  of  them  has  the  relation  to 
something  or  other;  and  that  the  codomain  of  R  is  the 
class  of  all  the  y's, — the  class  of  all  the  terms, — such  that, 
given  any  one  of  them,  something  or  other  has  the  relation 
to  it.  I  may  add  that  the  terms  in  the  domain  of  R  are 
often  called  the  referents  of  R  and  that  the  terms  in  the 
codomain  are  called  the  relata  of  R.  Some  relations  have 
fields;  others,  not.  R  has  a  field  if  and  only  if  the  domain 
and  codomain  are  of  the  same  type, — that  is,  are  com- 
posed of  individuals  or  else  of  classes  of  individuals  or 
else  of  classes  of  classes  of  individuals,  and  so  on, — and 
the  field  is,  if  there  be  one,  the  logical  sum  of  the  domain 
and  codomain, — the  class,  that  is,  containing  every  term 
in  the  domain  or  in  the  codomain  and  no  other  term. 
Thus,  if  <f>(x,  y)  be — x  is  a  husband  of  y — then,  if  yi  be  a 
wife  of  X\,  the  couple  (xi,  yi)  is  a  constituent  of  the  rela- 
tion; the  relation  "  husband  of"  is  the  class  of  all  such 
couples;  the  domain  is  the  class  of  husbands;  the  codo- 
main is  the  class  of  wives;  the  field  is  the  class  of  husbands 
and  wives;  each  husband  and  nothing  else  is  a  referent; 
each  wife  and  nothing  else  is  a  relatum;  observe  that  in 
this  example,  the  domain  and  codomain  have  no  common 
terms.  If  <j>(x,  y)  be — x  is  a  positive  integer  less  than  a 
positive  integer  y — then  the  relation  is  the  class  of  all 
couples  (xi,  yi)  such  that  xi  and  yi  are  positive  integers  of 
which  the  former  is  the  less;  every  integer  is  a  referent; 
every  integer  except  I  is  a  relatum;  and  so,  you  see,  the 
domain  includes  the  codomain,  but  the  converse  is  not 
true.  If  R  were  identity,  for  example,  or  equality  or 
diversity,  then,  as  you  easily  see,  the  domain  and  the 
codomain    would    each    include    the    other — they   would 


MORE  ABOUT  LIMITS  207 

coincide.  Of  relations  as  a  subject  I  have  already 
repeatedly  indicated  the  immensity  and  the  first-rate 
importance.  At  present,  I  am  asking  you  to  consider 
only  so  much  of  it  as  is  necessary  and  sufficient  for  our 
present  purpose,  which  is  that  of  preparing  us  to  under- 
stand certain  highly  important  meanings  of  the  term 
limit. 
Relations  are  endless  in  number  and  in  variety  and 
they  are  omnipresent  as  well  in  practical  life  as  in  abstract 
thought.  There  is  one  variety  (including  a  vast  multitude 
of  sub-varieties)  to  which  I  am  going  now  to  ask  your  best 
attention.  Before  defining  it,  it  will  be  helpful  to  con- 
sider a  simple  specific  example  of  it.  The  example  I  am 
going  to  use  is  one  of  the  relations  instanced  a  moment 
ago.  I  mean  the  relation  determined  by  the  propositional 
function :  x  is  a  positive  integer  less  than  a  positive  integer 
y.  Let  us  denote  the  relation  by  P.  Observe  what  P  is. 
It  is  the  class  of  couples:  (i,  2),  (1,  3),  (1,  4),  (1,  5),  .  .  .  ; 
(2,  3),  (2,  4),  (2,  5),  .  .  .  ;  (3,  4),  (3,  5),  ..;...;...; 
and  so  on  endlessly.  Note  that  P  has  a  field — the  class  of 
all  the  positive  integers.  The  relation  P  has  numerous 
properties;  let  me  ask  you  to  inspect  very  carefully  just 
three  of  them.  The  three  are  these:  (a)  if  n  be  in  P's 
field,  (w,  n)  is  not  a  constituent  of  P, — that  is,  nPn  is  a 
false  proposition, — that  is,  P  is  not  a  relation  which,  like 
identity,    holds   between    a   term    and   that   same   term; 

(b)  if  n  and  n'  are  in  P's  field,  then  either  (n,  n')  or  else 
(w',  n)  is  a  constituent  of  P — that  is,  nPn'  or  else  n'Pn\ 

(c)  \inPn'  and  n'Pn"y  then  also  nPn" .  Because  the  rela- 
tion has  these  three  properties,  it  is  called  a  serial  relation, 
or  a  series,  or  a  sequence,  or  a  specimen  of  linear  order. 
You  detect  at  once  how  to  define  these  equivalent  terms. 
The  definition  is  as  follows:  A  serial  relation  (or  series  or 


208  MATHEMATICAL   PHILOSOPHY 

sequence  or  linear  order)  is  a  relation  R  such  that:  (a)  if 
xRy,  x  and  y  are  not  the  same;  (b)  if  x  and  y  are  terms  in 
R's  field,  then  xify  or  else  yRx;  and  (c)  if  xRy  and  yi^z, 
then  xRz.  In  this  discussion,  let  us  use  the  shorter  names, 
sequence  and  series,  for  such  an  R,  instead  of  the  other 
names.  Evidently  the  above  P  is  a  specific  instance  of  a 
sequence  or  series.  Consider  another  instance,  say  P', 
where  P'  is  determined  by  the  prepositional  function: 
x  is  a  positive  integer  greater  than  a  positive  integer  y. 
You  see  that  P'  is  indeed  a  sequence.  Notice  that  P  and 
P'  are  different  sequences:  for  example,  the  couple  (i,  2) 
is  a  constituent  of  P  but  (2,  1)  is  not,  while  (2,  1)  is  a 
constituent  of  P'  but  (1,  2)  is  not.  Yet  the  field  of  P  is 
the  same  as  the  field  of  P' — namely,  the  class  of  positive 
integers.  You  readily  see  that,  if  the  field  F  of  a  given 
sequence  be  infinite,  there  are  infinitely  many  different 
sequences  having  F  for  their  field.  It  is  plain  that  the 
smallest  class  that  can  be  the  field  of  a  sequence  is  a  class 
having  two  and  only  two  members,  say,  a  and  b;  even 
in  this  case,  there  are  two  sequences  having  the  field  in 
common;  one  of  them  consists  of  the  couple  (a,  b),  the 
other  of  the  couple  (b,  a).  Let  me,  in  passing,  propose  an 
instructive  little  exercise.  Given  a  class  of  three  terms, 
a,  b  and  c,  show  that  there  are  six  sequences  having  the 
class  for  field,  that  each  sequence  is  a  class  of  three  couples, 
and  write  down  the  couples  for  each  case. 

In  our  introductory  study  of  sequences,  or  series,  it  is 
desirable  to  learn  something  more  of  the  subject's  lan- 
guage; for  as  supersimians,  we  must  chatter  about  the 
subject,  and  as  supersimian  philosophers,  we  must  try  to 
chatter  intelligibly.  If  the  relation  R  be  a  sequence  we 
say  that  the  referents  of  R  are  predecessors — predecessors 
for  R,  or  R  predecessors;    that  the  relata  of  R  are  sue- 


MORE   ABOUT   LIMITS  269 

cessors — successors  for  R,  or  R  successors;  that,  if  xRy, 
a:  is  a  predecessor  of  y  and  y  is  a  successor  of  x — more 
precisely,  that  x  is  an  R  predecessor  of  y  and  y  is  an  R 
successor  of  x.  You  see  immediately  that  every  term 
in  R's  domain  is  an  R  predecessor,  that  every  term  in 
R's  codomain  is  an  R  successor,  and  that  every  term  in 
R's  field  is  either  a  predecessor  or  a  successor  and  is 
generally  (not  always)  both.  Thus  in  the  case  of  our 
example  P,  I  is  a  predecessor  but  not  a  successor,  while 
every  other  integer  in  the  field  is  both;  on  the  other  hand, 
in  the  case  of  P',  I  is  a  successor  but  not  a  predecessor, 
while  every  other  positive  integer  is  again  both.  If  a 
term  t  be  an  R  predecessor  but  not  an  R  successor,  the 
sequence  R  is  commonly  and  conveniently  said  to  have  a 
beginning  t — to  begin  at  t;  thus  P  has  a  beginning,  it 
begins  at  i.  If  t  be  a  successor  but  not  a  predecessor,  the 
sequence  has  an  end  t — it  ends  at  t;  thus  P'  has  an  end, 
it  ends  at  I ;  P  is  endless,  P'  is  beginningless.  A  sequence 
may  have  both  beginning  and  end  or  neither.  An 
example  of  the  former  is  the  sequence  P"  determined  by 
the  propositional  function:  x  is  a  positive  integer  less  than 
a  positive  integer  y  not  greater  than  10;  you  see  that 
P"  is  a  sequence  and  that  it  has  a  beginning,  I,  and  an 
end,  10.  The  field  of  P"  is  finite.  Can  a  sequence 
whose  field  is  infinite  have  both  beginning  and  end?  Yes; 
consider  the  sequence  determined  by  the  propositional 
function:  *  is  a  real  number  (equal  to  or  greater  than  i) 
less  than  a  real  number  y  (not  greater  than  2);  you  see 
that  the  relation  determined  by  the  function  is  a  sequence, 
that  the  sequence  begins  at  1  and  ends  at  2,  and  that  the 
field  is  infinite — the  class  whose  terms  are  1,  2  and  all  the 
intervening  real  numbers.  For  an  example  of  a  sequence 
having   neither    beginning   nor   end,   we    may   take   the 


270  MATHEMATICAL    PHILOSOPHY 

series  P'"> — the  class  of  couples  (x,  y), — determined  by  the 
propositional  function:  x  is  a  real  number  (greater  than 
i)  less  than  a  real  number  y  (less  than  2);  it  is  clear  that 
P'"  is  beginningless  and  endless — each  term  in  its  field 
is  both  a  predecessor  and  a  successor.  If  t  be  a  term  in  the 
field  of  a  sequence  and  be  at  once  a  successor  of  t\  and  a 
predecessor  of  *2>  then  t  is  said  to  be  between  ti  and  t?.  If 
the  sequence  be  S,  we  may  say  that  the  terms  between 
ti  and  t2  are  S-intermediate  to  t\  and  tz.  If  between  every 
two  terms  in  the  field  of  a  sequence  there  is  a  term  of  the 
field,  the  sequence  is  said  to  be  dense;  thus,  P'",  for 
example,  is  dense,  while  P,  P\  P"  are  not.  You  will 
hardly  confuse  the  notion  of  a  dense  sequence  with  that 
of  a  dense  difference  kind  k.  The  amounts  of  such  a  kind 
constitute  a  field  of  a  dense  sequence  but  the  difference- 
kind  is  not  itself  a  sequence.  > 

It  is  noteworthy  that,  in  dealing  with  a  sequence, 
mathematicians  do  not  usually  state  explicitly  a  proposi- 
tional function  determining  it,  though  it  is  always  pos- 
sible and  often  helpful  to  do  so;  neither  do  they  usually 
indicate  explicitly  (as  above  done  in  the  case  of  P)  the 
class  of  couples  constituting  the  sequence,  though  this, 
too,  can  be  done  if  desired.  For  example,  if  a  mathema- 
tician wishes  to  invite  attention  to  our  sequence  P,  he 
will  ordinarily  say:    Consider  the  sequence 

1,  2,  3,  4,  ...  ,  n,  tt  +  i,  .  .  . 

He  will  probably  talk  as  if  the  row  of  numbers  were  the 
sequence,  though  it  is  not — the  sequence  being,  as  we  have 
seen,  a  certain  class  of  couples;  the  numbers  in  the  row 
constitute  the  field  of  P  but  the  field  as  such  he  will 
probably  not  mention;  he  will  speak  of  the  numbers  as 
the  terms  of  the  sequence,  though  they  are  merely  the 


MORE  ABOUT  LIMITS  271 

terms  of  the  field,  the  terms  of  the  sequence  (a  class  of 
couples)  being  the  couples  in  the  class;  and  he  will 
ordinarily  have  you  understand  that  a  number  on  the  left 
of  another  in  the  row  is  a  predecessor  of  the  latter  and  that 
the  latter  is  a  successor  of  the  former.  This  usual  row 
method  of  indicating  sequences  has  obvious  advantages, — 
it  is  mechanical,  spatial,  visual,  diagrammatic, — but  it 
has  to  be  used  with  care  if  confusion  and  error  are  to  be 
avoided,  for,  as  you  already  see  and  will  further  see,  it 
disguises  some  of  the  nicer  logicalities  involved.  For 
example,  the  mathemetician  may  indicate  the  sequence 
P'  (instead  of  P)  by  writing  the  foregoing  row  of  numbers; 
in  this  case,  a  number  to  be  a  predecessor  of  another 
must  be  on  the  right  of  the  latter  instead  of  on  the  left  of 
it;  you  see  that  the  notion  left-right  (or  its  like)  is  not 
that  of  predecessor-successor;  the  former  is  spatial  and 
sensuous,  the  latter  logical  and  supersensuous;  in  the  case 
of  P,  3  is  the  predecessor  of  4,  not  because  3  is  on  the  left 
of  4  in  the  row,  but  simply  because  3P4;  and  in  the  case 
of  P\  4  is  the  predecessor  of  3,  not  because  the  former  is 
on  the  latter's  right,  but  because  \P'^ ;  again  P  begins  at 
1,  not  because  1  begins  the  row  of  numbers,  for,  you  see, 
P'  ends  at  1,  despite  the  fact  that  the  row  begins  at  I. 
Keeping  such  precautions  in  mind,  we  may  often  very 
conveniently  employ  the  row  method  of  indicating  or 
representing  sequences. 

We  are  at  length  almost  prepared  for  a  certain  new 
definition  of  the  term  "  limit  ";  but  there  remains  to  be 
explained  one  further  preliminary.  It  is  a  sfollows:  If 
xRy  implies  xR'y,  the  relation  R  is  said  to  be  included 
in  the  relation  R';  in  other  words,  R  is  included  in  R'  if 
every  couple  (xy  y)  in  the  class  of  couples  constituting  R 
is  a  couple  in  the  class  of  couples  constituting  R';    if  R 


272  MATHEMATICAL    PHILOSOPHY 

be  included  in  R'  we  say  naturally  that  R  is  a  part  of  R' . 
I  need  hardly  point  out  the  fact  that  a  relation  includes 
itself  and  is  thus  a  part  of  itself.  R  and  R'  are  identical 
when  and  only  when  each  is  a  part  of  the  other.  Now 
suppose  R  to  be  a  part  of  R'  and  suppose  R'  to  be  a 
sequence  (series);  then  R  is  also  a  sequence  obviously; 
and  thus,  as  you  see,  one  sequence  may  be  a  part  of 
another.  It  is  plain  that,  if  a  sequence  R  be  a  part  of  a 
sequence  Rf,  the  field  F  of  i?  is  a  part  of  the  field  F'  of  i?'; 
and  conversely  that,  if  a  class  F  (of  two  or  more  terms) 
be  a  part  of  the  field  F'  of  a  sequence  R',  then  .F  is  the 
field  of  a  sequence  i?  included  in  R'.  Consider,  for 
example,  our  familiar  friend  P;  its  F  is  the  class  of 
positive  integers;  take  any  part  of  F, — any  part  contain- 
ing two  or  more  terms, — say,  the  class  C  composed  of 
2,  3  and  7;  C  is  the  field  of  the  sequence  composed  of  the 
couples  (2,  3),  (2,  7),  (3,  7);  this  sequence  is  a  part  of  P. 
It  will  be  enlightening  to  notice  that  P  is  itself  a  part  of 
another  sequence;  let  0  be  the  sequence  determined  by 
the  propositional  function,  a;  is  a  positive  real  number  less 
than  a  positive  real  number  y;  you  see  that  Q  is  a  sequence, 
that  its  field  is  the  class  of  all  positive  real  numbers,  that 
this  field  includes  the  field  of  P;  that  every  couple  in  P 
is  also  in  Q;  and  that  P  is  a  part  of  Q.  You  readily  see 
that  if  one  sequence  be  a  part  of  a  second,  and  the 
second  a  part  of  a  third,  the  first  is  a  part  of  the  third. 

When  we  speak  of  the  (amount  of)  difference  between 
a  term  t  in  the  field  of  a  sequence  S  and  a  term  /',  let  it  be 
always  understood  that  t'  is  either  in  the  field  of  S  or  in 
the  field  of  a  sequence  including  S. 

I  hope  we  are  now  prepared  to  grasp  the  following 
definition  of  the  term  "  limit." 

D3:   Let  V  be  a  variable  whose  range  R  is  included  in 


MORE  ABOUT  LIMITS  273 

the  field  F  of  a  sequence  S,  and  (as  before)  let  k  be  an 
available  difference-kind;  if  there  be  in  F  a  term  t  such 
that,  however  small  a  ^-neighborhood  of  t  be  chosen, 
there  is  in  the  neighborhood  an  R  term  t'  differing  by 
more  than  null  from  t  and  being  such  that  all  R  terms 
between  t  and  t'  are  in  the  neighborhood,  then  t  is  an  S 
&-limit  of  V. 

The  meaning  of  D%,  which  is  a  bit  subtle  and  sly,  may 
be  made  evident  by  means  of  a  few  examples.  In  adduc- 
ing examples  it  will  be  convenient  to  make  some  use 
of  the  customary  row  method  of  representing  sequences. 

For  a  simple  example,  let  S  be  the  sequence  determined 
by  the  propositional  function:  #  is  a  fraction  (having  i 
for  numerator  and  a  positive  integer  for  denominator) 
greater  than  a  fraction  y  (having  I  or  zero  for  numerator 
and  a  positive  integer  for  denominator).  F  is  composed 
of  the  numbers  in  the  row 

h  I)  i  i  •  •  •  (ad  infinitum),  0. 

Let  V  be  the  variable  whose  range  R  is  the  class  of  all  the 
F  terms  except  0;  let  k  be  the  kind  of  difference  in  respect 
of  which  we  compare  the  values  or  magnitudes  of  rational 
fractions  (as  when  we  say  \—  $  =  £).  The  question 
is:  Has  V  an  S  £-limit  t\  The  answer  is  yes:  t  is 
such  a  limit  if  t  be  zero  (0).  To  prove  it,  suppose  chosen  a 
^-neighborhood  d  of  0,  however  small;  there  is  no  restric- 
tion upon  the  choice  of  d  save  that  d  must  be  a  positive 
rational  number;  it  is  plain  that  there  is  in  R  a  number  t 
differing  from  0  by  more  than  null  (zero) and  by  less  than 
d;  it  is  evident  that  such  a  /  and  all  the  R  terms  between 
t  and  0  are  in  the  chosen  neighborhood;  and  hence  0 
is,  as  said,  an  S  &-limit  of  V. 

It  should  be  said  in  passing  that  a  V  having,  under 


274  MATHEMATICAL   PHILOSOPHY 

Z>3,  zero  (or  null)  for  limit  is  called  an  infinitesimal — a 
term  of  great  importance  in  most  branches  of  mathe- 
matics.    We  will  return  to  it  if  time  permits. 

Let  us  choose  another  difference-kind  and  see  what 
happens  when  V,  F  and  S  have  the  same  meanings  as 
above.  Observe  that  the  denominators  of  the  F  terms  are 
positive  integers,  for  0  may  be  written  0/n  where  n  is  a 
positive  integer.  We  may  compare  the  F  terms  with  sole 
reference  to  the  values  of  their  denominators — with 
reference,  that  is,  to  the  difference-kind  k'  in  respect  of 
which  we  compare  the  magnitudes  of  positive  integers 
as  such.  Of  k'  the  only  amounts  d  are:  null,  I,  2,  3,  4,  ...  ; 
hence  the  smallest  ^'-neighborhood  d  of  0  is  that  for  which 
d=i;  as  no  R  term  differs  from  0  (0/n)  by  an  amount 
of  difference-kind  k'  more  than  null  and  less  than  1,  it  is 
seen  that  no  R  term  t'  differing  by  more  than  null  of  the 
difference-kind  k'  from  0  is  in  the  neighborhood  of  0  for 
which  d  =  i;  therefore,  0  is  not  an  S  £'-limit  of  V.  And 
so  is  justified  the  mention  of  k  in  Z)3. 

We  have  just  seen  that,  though  a  t  be  an  S  &-limit  of  V, 
it  may  not  be  an  S  &'-limit  of  V  if  k  and  k'  be  not  the 
same.  We  may  now  show  that,  though  a  t  be  an  S  &-limit 
of  V,  it  may  not  be  an  S'  £-limit  of  V,  if  S  and  S'  be 
different  sequences.     Consider  the  numbers  in  the  row 

!>  3>  i>  •  •  •  (&d  infinitum),  t,  0. 

Let  S'  be  a  sequence  having  the  class  of  thes°  numbers 
for  its  field  and  let  S'  be  such  that,  if  a  is  an  S'  predecessor 
of  b,  then  a  and  b  are  in  the  row,  a  on  the  left  of  b,  and 
that  any  number  in  the  row  is  an  S'  predecessor  of  all  the 
numbers  on  its  right  and  an  S'  successor  of  all  the  numbers 
on  its  left.     S',  as  you  see,  is  now  completely  determined — 


MORE  ABOUT  LIMITS  275 

we  know  all  the  couples  constituting  it;  and  you  note  that 
its  field  F  is  the  same  as  that  of  S  above.  Let  V  be  the 
same  as  before  and  let  k  be  the  same  as  in  the  above 
paragraph,  in  which  0  was  found  to  be  an  S  £-limit  of  V . 
The  question  is:  is  0  an  Sr  &-limit  of  V\  The  answer  is 
no,  as  you  readily  see;  for  choose  a  ^-neighborhood  d  of  0, 
where  d  is,  say,  \ ;  any  F  term  /'  differing  by  more  than 
null  (zero)  from  0,  if  it  be  in  the  chosen  neighborhood,  is, 
as  you  see,  a  predecessor  of  |;  and  so  y>  though  it  is  be- 
tween such  t'  and  0,  is  not  in  the  chosen  neighborhood; 
accordingly,  as  said,  0  is  not  an  S'  &-limit  of  V . 

Comparing  D2  with  D3  you  observe  that  Dz  contem- 
plates F's  range  as  a  part  of  the  field  of  a  sequence  and 
that  D>>  does  not;  you  notice,  too,  that  D3  contains  the 
same  conditions  as  D2  contains  and  one  other — the 
"  between  "  condition  (which  would  indeed  be  meaningless 
in  D>  inasmuch  as  D2  does  not  regard  V\  range  as  included 
in  the  field  of  a  sequence).  It  follows  that  if  a  V  have  a 
limit  t  under  D%,  the  same  V  has  t  for  limit  under  D2. 
Is  the  converse  true?  It  is  easy  and  instructive  to  show 
by  an  example  that  it  is  not.  Consider  the  numbers  in 
the  row 

(i+i),  (i-i),  (i+i),  (tV-0,  (A+i),  GrV-i),  ...; 

which  are  the  same  as  the  numbers  in  the  row 

t3       »  J-5.     .2-3.  63 

>    —  T,  Iff,    ~  TS,  "3"2",    —  FT>   •   •   •    > 

let  S  be  a  sequence  such  that,  if  aSb,  a  and  b  are  in  the  row, 
a  on  the  left  of  b,  and  that  each  number  in  the  row  is  an 
S  predecessor  of  every  number  on  its  right.  S's  field  F 
is  the  class  of  the  numbers  in  the  row.     Let  k  be  the 


276  MATHEMATICAL    PHILOSOPHY 

difference-kind  appropriate  for  comparing  (as  by  sub- 
traction) the  values  of  real  numbers.  Let  V  be  the 
variable  whose  range  is  F.  It  is  easy  to  see  that  I  and  —  I 
are  both  of  them  limits  of  V  under  Az  and  that  neither  of 
them  is  an  S  limit  of  V  under  D3.  For  choose  a  neighbor- 
hood d  of  1,  no  matter  how  small;  plainly  there  is  an  F 
term  (a  positive  number  in  the  row,  but  not  a  negative 
one)  differing  from  1  numerically  by  less  than  d,  and 
such  a  term  is  in  the  chosen  neighborhood;  accordingly 
1  is,  by  Z>2,  a  limit  of  V;  but  1  is  not  an  S  limit  of  V  for  1 
is  not  even  a  term  of  S's  field.  Like  reasoning  would  show 
that  —1  is  a  limit  of  V  under  D2  but  is  not,  under  D3, 
an  S  limit  of  V,  where  S  is  the  sequence  above  indicated. 

We  have  seen  that,  if  two  sequences  S  and  S'  have  a 
common  field  F  and  if  V  be  a  variable  whose  range  R 
is  a  part  of  F,  a  term  t  may  be  an  S  limit  of  V  without 
being  an  Sf  limit  of  V.  This  fact  is  so  important  that  it 
seems  advisable  to  give  it  further  exemplification.  Let  F 
be  the  class  of  all  the  positive  rational  numbers  and  zero. 
Consider  the  following  sequences  Si,  S2,  S3,  having  F 
for  their  common  field. 

Si  is  to  be  such  that,  if  xSiy,  x  and  y  are  in  F  and  x 
is  less  than  y;  and  such  that,  if  x  and  y  be  in  F,  then 
xSiy  if  and  only  if  x  is  less  than  y.  We  commonly  say 
that  Si  as  defined  arranges  the  terms  of  F  in  the  order 
of  increasing  magnitude. 

To  define  S2  consider  the  row 

/•„\      1       1      1  .      f»      2       2       2  .       3      3      3 

\d)    T,  ?Z,  ^">  •  •  •   5     U,  T,  "3",  Z,  ■   •   •    >     1,   2,   4,   •••    ,     •  •   •    j     •   •   • 

You  observe  that  the  row  contains  all  and  only  the  terms 
of  F;  S2  is  to  be  such  that,  if  xS2y,  x  is  on  the  left  of  y  in 
the  row,  and  that  any  term  in  the  row  is  an  S2  predecessor 
of  every  number  on  its  right. 


MORE  ABOUT  LIMITS 

To  define  S3  consider  the  array. 


277 


Fig.  27. 

A  little  inspection  shows  that  the  array  contains  all  and 
only  the  terms  of  Fy  zero  excepted.  The  arrows  indicate 
how  the  terms  of  F  may  be  arranged  as  in  the  row 


(U\      1213214321 

\0)    ±,  T,  U,  T,  "J,  S,  T,  IT,  ?,  T, 


;  0. 


Notice  that  the  scheme  gives  each  F  term  a  definite  place 
in  the  row.  S3  is  to  be  such  that,  if  xS^y,  x  is  on  the  left 
of  y  in  (b)  and  that  every  term  in  (b)  is  an  S3  predecessor 
of  every  term  on  its  right. 

Now  let  V  be  the  variable  having  for  its  range  the 
class  of  all  F  terms  except  zero,  and  let  k  be  the  familiar 
difference-kind  we  have  in  mind  when  we  say  the  (amount 
of)  difference  between  this  real  number  and  that  is  such- 
and-such.     Applying  Dz  to  V  you  will  readily  find  that 


278  MATHEMATICAL    PHILOSOPHY 

zero  and  all  positive  rational  numbers  are  Si  limits  of  V\ 
that  zero  and  nothing  else  is  an  S2  limit  of  V\  and  that 
V  has  no  S3  limit  whatever.  Two  additional  facts  are 
worth  noting  here:  one  of  them  is  that,  under  D2  zero 
and  every  positive  real  number  is  a  limit  of  V;  the  other 
is  that,  under  D3,  zero  and  every  positive  real  number  will 
be  an  S  limit  of  F,  if  S's  field  be  the  class  of  positive  reals 
and  zero,  and  if  S  arrange  the  terms  of  F  in  the  order  of 
increasing  magnitude. 

I  shall  leave  it  to  you  to  practise  to  your  heart's  con- 
tent in  applying  D3  to  such  various  variables  and  sequences 
as  you  can  readily  find  or  devise. 

Presently,  I  shall  ask  you  to  consider  a  fourth  concep- 
tion of  limit.  Before  doing  so,  I  wish  to  call  your  atten- 
tion to  a  curious  nice  little  dispute  that  now  and  then 
arises  respecting  the  notion  of  limit  as  defined  by  D% 
or  by  a  virtual  equivalent  of  D3.  The  dispute  arises  out 
of  confusion  due  partly  to  the  row  method  of  indicating 
sequences  and  partly  to  the  custom  of  speaking  figura- 
tively of  a  variable  as  if  it  actually  changed,  varied, 
increased,  decreased,  and  so  on,  instead  of  merely  repre- 
senting "  any  one  "  of  the  terms  of  some  specified  class. 
I  can  best  present  the  matter  by  means  of  an  example 
or  two.  Consider  the  three  sequences  indicated  by  the 
three  number  rows. 

/_   \    ,    1      3      7      15  .T 

\T\)     •    ~2,    4,     8,     16,    •    •    •     y    1 

/.,    \     .     3       5       9       17  ,T 

(To)     •    2,    T,    8,    H,    •    •    •    J    I 

/„    \     .     1       3       3       5  .    T 

l/3j     •    2)    S)   I)    I)        .    .    .     ,    I 

You  note  that  the  three  sequences  are  distinct  and  that 
their  fields  are  distinct.  If  the  ranges  of  the  variables 
V\,  Vi->  Vz  contain  respectively  the  same  terms  as  the 
fields  except  1,  you  easily  see  that,  under  D3,  1  is  a  limit 


MORE   ABOUT  LIMITS  279 

of  V\,  of  V2  and  of  V?,.  It  is  customary  to  say,  "  As  V\ 
runs  along  the  sequence  (n)  from  left  to  right  it  approaches 
1  as  its  limit  "  or  to  use  some  equivalent  equally  figurative 
speech;  and  similarly  for  V2  and  Vz.  It  is  noticed  that 
the  V\  in  running  along  their  so-called  sequences  get 
nearer  and  nearer  to  their  limits  but  never  reach  them. 
The  question  arises:  Is  it  possible  for  a  sequence  having  a 
limit  to  be  such  that  the  variable,  in  the  course  of  its 
approaching  the  limit,  reaches  it  one  or  more  times? 
Some  say  no;  others  say  yes.  The  latter  attempt  to  justify 
their  answer  substantially  as  follows:  Consider,  they  will 
say,  the  sequence 

(r)  i,  I,  1,  f,  f,  1,  i,  I,  1,  •  •  .  (ad  infinitum);  1, 

got  from  (r^)  by  inserting  1  after  each  of  the  successive 
pairs  of  numbers  in  (73) ;  observe,  they  will  say,  that  if  a  V 
runs  along  (r),  skipping  the  third  term,  the  sixth  and  so 
on,  it  will  approach  the  same  limit  (namely,  1)  as  if  it 
ran  along  (r^),  and  that,  if  it  runs  along  (r)  without 
skipping,  it  will  again  evidently  approach  the  same  limit, 
1,  but  in  this  case  will  actually  reach  1  infinitely  often  in 
endlessly  approaching  it;  and  so  you  are  expected  to  see 
not  only  that  (r)  is  a  sequence  having  a  limit  but  that, 
while  endlessly  approaching  it,  it  actually  reaches  it  again 
and  again  and  again.  You  instinctively  feel  that  you  are 
being  hocus-pocused  by  such  argument,  and  your  instinct 
is  sound.  What  is  the  trick?  It  is  easy  to  detect.  The 
juggler  (we  may  call  him  a  juggler,  though  he  does  not 
intend  to  deceive)  asks  us  to  regard  (r)  as  a  sequence  or 
at  all  events  as  indicating  a  sequence.  Let  us  try  to  do 
so  in  good  faith.  If  (r)  be  or  indicate  a  sequence  S,  what  is 
the  field  F?  The  answer  is  obvious:  the  terms  of  F  are 
the  numbers  in  (r3).     Among  these  is  1,  the  final  number 


S80  MATHEMATICAL    PHILOSOPHY 

of  (f3)  and  of  (r).  But  (r)  indicates  that  F  contains  a 
host  of  i's — the  "  inserted  "  i's;  but  if  these  are  in  F, 
they  are  S  predecessors  and  successors,  and  we  have  1S1 
contrary  to  the  definition  of  a  sequence.  You  see  that 
(r)  neither  is,  nor  property  indicates,  a  sequence.  (It  is 
of  course  possible  to  define  the  terms  "  sequence  "  and 
"  limit  "  so  that  a  sequence  having  a  limit  may  be  such 
that  the  variable  in  running  towards  the  limit  reaches  it 
one  or  more  times.)  Here  is  a  good  place  to  emphasize 
the  fact  that  the  field  of  a  sequence  never  contains  two 
identical  terms.  Why  not?  Because  a  field  is  a  class, 
and  a  class  contains  all  and  only  the  verifiers  of  some 
propositional  function,  say,  <f>(x);  if  x\  be  a  verifier  of 
4>(x),  then  x\  is  a  term  or  member  of  the  class;  it  is 
evident  that  as  such  a  member,  it  occurs  but  once.  We 
do  indeed  often  speak  (unprecisely)  as  if  such  were  not  the 
case;  but  when  we  speak  of  a  and  a!  as  being  identical 
members  of  a  class,  we  mean  that  a  and  a'  are  two  dif- 
ferent symbols  for  one  and  the  same  member  of  the  class 
and  we  do  not  mean  that  the  two  symbols  are  themselves 
members  of  the  class. 

Serial  (Ordinal)  Definition  of  the  Term  "  Limit." — We 
have  now  before  us  three  definitions — £>i,  A2,  Dz — of  the 
term.  It  is  important  to  observe  that  each  of  them 
essentially  involves  the  notion  of  quantity;  they  involve  it, 
for  they  involve  the  notion  of  the  neighborhood  of  a  term, 
and  this  notion  is  quantitative;  a  given  neighborhood 
has  a  size;  another  one  is  larger  or  smaller;  neighbor- 
hoods are  among  the  things  differing  from  one  another  in 
respect  of  magnitude — quantity  is  of  their  essence.  We 
should  not  fail  to  observe,  too,  that,  while  the  three 
definitions  thus  agree  in  involving  the  notion  of  quan- 
tity, Z>3  involves  also  the  notion  of  a  sequence  or  series, — 


MORE  ABOUT  LIMITS  281 

a  non-quantitative,  purely  ordinal  notion, — and  that  Di 
and  Di  do  not.  I  mean  that  D3  contemplates  the  vari- 
able's range  as  being  the  field  (or  a  part  of  the  field)  of  a 
series,  or  sequence,  and  that  Di  and  D2  do  not.  As 
ontologists  you  may  no  doubt  contend  that  the  terms  of 
any  given  class  and  hence  the  terms  of  any  given  variable's 
range  are,  quite  independently  of  our  intention  or  will, 
arranged  once  for  all  and  eternally  in  every  variety  of 
sequence  of  which  they  are  capable.  I  am  not  disputing 
the  justice  of  that  contention;  conceding  it  to  be  just, 
granting  the  eternal  existence  of  all  the  sequences  possible 
for  a  given  range,  I  am  merely  signalizing  the  fact  that  Di 
and  Z>2  disregard  them  each  and  all,  and  that  D%  does  not; 
D3  regards  the  variable's  range  as  an  ordered  class  of 
terms;  D\  and  D2,  disregarding  order,  regard  the  vari- 
able's range  as  an  orderless  collection.  We  may  say, 
then,  that  D\  and  D2  are  quantitative  definitions  and  that 
Di  is  mixed — both  quantitative  and  serial. 

It  is  natural  to  ask  whether  the  term  "  limit  "  some- 
times denotes  a  purely  serial  conception.  The  answer  is 
affirmative.  The  following  definition  presents  such  a 
definition  of  the  term. 

D4:  Let  V  be  a  variable  whose  range  R  is  included  in 
the  field  F  of  a  sequence  (series)  S;  if  an  F  term  /  be  such 
that,  given  any  S  predecessor  t'  of  t  among  the  R  terms, 
there  is  an  R  term  between  t'  and  t,  or  such  that,  given 
any  S  successor  /'  of  /  among  the  R  terms,  there  is  an 
R  term  between  t  and  t',  then  t  is  an  S  limit  of  V. 

Let  us  at  once  cite  some  simple  examples.  Consider 
the  sequence. 

Si  :  1,  2,  3,  4,  .  .  .  (ad  infinitum),  £,  |. 
Let  predecessor-successor  mean  left-right;    let  the    terms 


282  MATHEMATICAL    PHILOSOPHY 

of  F  be  the  numbers  in  the  row;  let  the  terms  of  F's 
range  jR  be  the  row's  integers.  You  see  that  |  is  an  Si 
limit  of  V  and  that  \  is  not.  Why  not?  Notice  that  \ 
is  not  an  Si  limit  of  V  under  Dz  nor  a  limit  of  V  under 
D2.  If  a  term  be  a  limit  of  a  V  under  D3  or  D2,  must  it 
be  a  limit  under  Z>4?  If,  in  the  foregoing  example,  we 
suppose  the  R  to  include  |  (besides  the  integers),  will  the 
new  V  have  an  S  limit?  Why  not?  If  we  strike  out 
\  and  \  and  suppose  R  to  coincide  with  the  new  F  (of  the 
new  Si),  will  V  have  an  Si  limit?     Why  not? 


Fig.  28. 

For  other  examples  consider  the  following.  Let  S2 
be  a  sequence  having  for  its  F  the  points  of  the  line 
segment  AD,  including  A  and  D.  Let  us  take  predecessor- 
successor  to  mean,  as  before,  left-right;  let  the  R  of  V 
be  composed  of  A  and  the  other  points  preceding  B;  you 
see  that  all  the  points  in  R  and  the  point  B  but  no  other 
F  points  are  S2  limits  of  V.  Notice  that  the  same  would 
be  true  if  we  supposed  R  to  include  B.  Suppose  the  F 
of  S2  to  be  composed  of  B,  B's  predecessors,  C  and  C's 
successors;  and,  as  before,  let  the  terms  of  R  be  B's 
predecessors;  you  readily  see  that  the  S2  limits  of  V  are 
B,  B's  predecessors  and  C;  that  all  of  these  except  C  are 
S2  limits  of  V  under  D3  and  limits  of  V  under  D2;  and 
that,  under  D4,  C  is  not  an  S2  limit  if  R  include  B  (as  well 
as  its  predecessors).  Why  not?  Next  suppose  the  F  of 
S2  to  consist  of  all  the  points  of  AD  except  B  and  C,  and 
let  the  terms  of  R  be  the  midpoint  A'  of  AB,  the  midpoint 
of  A'B,  and  so  on,  the  midpoint  B'  of  BC,  the  midpoint  of 
B'C,  and  so  on,  the  midpoint  C  of  CD,  the  midpoint  of 
CD,  and  so  so;  then  show  that  D  is  the  only  S2  limit  of 


MORE   ABOUT  LIMITS  283 

V  under  either  At  or  D3  but  that,  under  Do,  B  and  C  as  well 
as  D  are  limits  of  V. 

With  the  foregoing  non-quantitative — purely  serial — 
conception  of  limit,  you  can  make  yourselves  familiar 
by  applying  the  definition  to  numerous  examples  which 
you  can  readily  construct  or  easily  find,  for  they  abound 
on  every  hand. 

I  have  now  spoken  of  limits  and  limit  conceptions  at 
far  greater  length  than  I  had  originally  intended  to  do. 
If  I  have  thus  exhausted  your  interest  and  patience,  I 
assure  you  that  I  have  by  no  means  exhausted  the  subject. 
There  are  in  use  yet  other  conceptions  of  the  term  limit 
and  connected  therewith  many  interesting  and  important 
refinements, — refinements  of  refinements,— with  which, 
however,  I  do  not  intend  to  trouble  you.  There  remain 
two  questions  which  must  have  occurred  to  you  and 
which  I  am  sure  you  will  desire  to  consider  before  we 
take  final  leave  of  the  subject.  One  of  them  is  easy  and 
admits  of  a  brief  answer.  The  question  is:  In  view  of  the 
variety  of  senses  in  which  mathematicians  employ  the 
term  "  limit,"  how  do  they  manage,  if  they  do  manage, 
to  avoid  confusion — confusion  of  themselves  and  others? 
The  answer  is:  They  do  not  always  avoid  it,  but  in  general 
they  do,  and  they  do  so,  as  I  have  already  intimated,  by 
indicating  either  explicitly  or  contextually,  when  speaking 
of  a  limit,  the  sense  in  which  the  term  is  to  be  understood. 
The  second  question  relates  to  the  scientific  and  philo- 
sophic importance  of  the  term.  Both  by  dwelling  on  it  so 
long  and  by  explicit  statement,  I  have  said  that  its  impor- 
tance is  very  great.  I  wish  now  to  show  that  the  estimate 
is  just  and  how  it  is  so. 

Scientific  and  Philosophic  Importance  of  the  Term 
Limit. — As  to  its  scientific  importance,  the  task  of  show- 


284  MATHEMATICAL   PHILOSOPHY 

ing  it  is  very  easy  if  we  take  "  scientific  "  in  its  stricter 
and  narrower  sense.  We  may  go  at  once  to  the  heart  of 
the  matter  by  reflecting  a  little  upon  the  most  rigorously 
scientific  of  scientific  subjects  and  procedures — the  Dif- 
ferential and  Integral  Calculus — and  upon  its  ramifications 
and  its  applications.  Some  of  you  have  had  a  beginner's 
course  in  the  calculus;  others  of  you,  not;  I  am  not 
going  to  offer  here  an  introduction  to  it  but  will  merely 
state  succinctly,  by  way  of  reminder  or  of  information,  a 
few  such  facts  respecting  it  as  will  make  indubitable  the 
great  scientific  importance  of  the  term  in  question.  One 
of  the  facts  is  that  the  Calculus  is  primarily  and  mainly 
concerned  with  what  mathematicians  call  continuous 
functions  (or  variables),  and  that  both  functional  con- 
tinuity and  functional  discontinuity,  with  which  latter 
the  calculus  is  also  concerned,  are  not  only  defined  by 
means  of  limits,  but  are  indeed  not  otherwise  definable. 
Another  of  the  salient  facts  is  that  among  the  host  of 
ideas  met  with  in  the  Calculus  three  ideas  are  supreme — 
namely,  those  denoted  by  the  terms  Derivative,  Anti- 
derivative  (or  Indefinite  Integral)  and  Definite  Integral — 
and  the  three  essentially  involve  a  limit  conception,  the 
first  and  third  of  them  directly,  the  second  one  indirectly. 
It  follows,  as  you  see,  that  in  all  the  multifarious  ramifica- 
tions and  applications  of  the  Calculus,  whether  in  differen- 
tial equations  or  function  theories  or  geometry  or  mechan- 
ics or  astronomy  or  physics  or  chemistry  or  other  fields 
into  which  the  calculus  has  found  or  is  inevitably  finding 
its  way,  some  variety  of  limit  conception  is  continually 
playing  an  indispensable  scientific  role.  Indeed  it  is  only 
by  prolonged  meditation  upon  the  matter  that  one  can 
even  fairly  begin  to  realize  how  very  deeply  the  progress 
of  science   and   therewith   of  civilization   depends   upon 


MORE  ABOUT  LIMITS  285 

ideas  denoted  by  the  modest  little  five-lettered  word — 
limit. 

What,  broadly  speaking,  we  may  call  its  philosophic 
significance  is  less  well  understood  for  the  reason  that  it 
has  been  neglected.  It  has  been  neglected  because  but 
few  mathematicians  have  been  interested  in  it  and  but 
few  philosophers  have  been  mathematically  qualified  to 
treat  it.  If  only  the  concept  of  limit  and  the  role  thereof 
had  been  familiar  in  the  days  of  Plato!  How  it  would 
have  enriched  and  fortified  his  dialectic.  In  his  hands 
the  concept  would  have  been  a  new  spiritual  instrument 
of  immeasurable  power;  in  his  thought  it  would  have 
opened  new  ways  to  the  inner  vision  of  supernal  light; 
in  his  brightest  pages  it  would  have  been  the  secret  and 
source  of  a  yet  stranger  and  brighter  glory.  His  shining 
Absolutes, — absolute  justice,  absolute  beauty,  absolute 
truth,  absolute  good, — whose  "  perception  by  pure  intelli- 
gence "  brings  us,  said  he,  "  to  the  end  of  the  intellectual 
world,"  would  not  have  appeared  as  ends,  or  final  terms, 
of  any  sequences  or  progressions  in  the  intellectual  world 
nor  even  as  limits  of  such  progressions  but,  as  I  intimated 
in  the  initial  lecture  of  the  course,  the  absolutes  would 
have  appeared  as  supernal  ideals,  over  and  above  every 
type  of  excellence  in  which  intellectual  progress  is  pos- 
sible. And  thus  the  Platonic  philosophy  would  have 
advanced,  in  even  greater  measure  than  it  did  advance, 
the  science  of  Idealization — the  science,  I  mean,  which 
has  for  its  appropriate  subject-matter  those  spiritual 
phenomena  of  life  which  the  terms,  ideal  and  idealization, 
rightly  understood,  denote.  In  saying  this,  I  have  in 
some  measure  anticipated  the  outcome  of  considerations 
not  yet  adduced,  and  so  I  must  ask  you  to  reserve  your 
judgment  for  a  little  time. 


286  MATHEMATICAL    PHILOSOPHY 

In  order  to  arrive  at  a  fair  estimate  of  the  philosophic 
significance  of  limit  concepts  and  limit  processes, — in 
order,  that  is,  to  win  a  fair  sense  of  their  function  and 
service  in  the  life  of  Thought  taken  in  all  its  varieties  and 
scope, — it  is  necessary  as  a  preparation  to  examine  the 
matter  a  little  further  in  mathematical  light  for  it  is 
here  and  not  elsewhere  that  concepts  of  limit  and  limit 
processes  are  seen,  and  seen  at  work,  in  their  nakedness 
and  purity.  As  beheld  in  that  light,  conceptions  of  limit, 
apart  from  any  question  regarding  their  instrumental 
value,  are  objects  of  no  little  interest — a  fact  well  worthy 
of  passing  mention,  though  I  do  not  insist  upon  it  in  this 
connection.  Regarding  instrumental  value,  we  have  seen 
that  limit  concepts  enable  us  to  discriminate  and  classify 
variables  with  reference  to  the  constitution  of  their 
ranges  and  to  the  connections  of  these  with  series;  we 
have  seen  that  limit  concepts  are  essential  to  the  formation 
and  so  to  the  meanings  of  innumerable  other  concepts, 
many  of  them  of  great  import,  as  that  of  functional  con- 
tinuity or  that  of  derivative,  instanced  a  moment  since; 
you  know,  or  (if  not)  you  can  quickly  learn  by  glancing 
at  mathematical  literature  that  limit  concepts  play  an 
indispensable,  perhaps  the  chief,  role  in  the  conduct  of 
proofs,  or  demonstrations,  in  all  branches  of  Analysis  and 
its  applications.  I  wish  now  to  invite  your  best  attention 
to  the  fact  that,  over  and  above  the  foregoing  types  of 
service,  limit  concepts  render  an  invaluable  service  of  a 
radically  distinct  kind  in  connection  with  that  very 
familiar  yet  always  strange  thing  which  we  are  wont  to 
call  "  generalization."  I  mean  the  kind  of  generalization 
which  consists  in  our  somehow  contriving  so  to  extend  the 
meaning  of  an  established  concept  as  to  bring  within  its 
enlarged  scope, — as  under  the  unity  and  order  of  a  new 


MORE  ABOUT  LIMITS  287 

empire, — what  had  been  seemingly  unconnected,  recip- 
rocally alien  provinces  of  thought. 

The  meaning  and  justice  of  what  I  have  just  now  said 
may  be  made  evident  by  means  of  simple  examples. 
Three  or  four  little  ones  will  suffice,  and  we  can  both 
shorten  the  task  and  enliven  it  by  speaking  of  our  vari- 
ables in  the  customary  dynamic  fashion. 

In  the  first  example,  I  am  going  to  ask  you  to  imagine 
that  we  have  arrived  at  a  stage  of  mathematical  evolution 
where  we  are  familiar  with  the  ordinary  fractions,  or 
ratios,  including  such  as  1,  f ,  .  .  .  ,  which  for  convenience 
we  will  write  1,2,...  ;  and  that  we  know  nothing  of  so- 
called  irrational  numbers.  Let  S  be  the  sequence  of  the 
ratios  arranged  in  the  natural  order  of  increasing  magni- 
tude. Let  Vx  represent  any  ratio  less  than  2  (i.e.,  f) 
and  let  V2  represent  any  one  greater  than  2.  You  imme- 
diately see  that,  under  either  D3  or  D4,  2  is  a  limit  of  Vx 
and  also  of  /£,  Vx  approaching  it  from  below  and  V2 
from  above.  Observe  that  neither  of  the  F's  can  reach 
the  limit;  one  of  them  is  always  less,  the  other  always 
greater,  than  2;  they  can,  however,  so  close  in  upon  2 
as  to  make  the  difference  between  them  less  than  any 
preassigned  positive  ratio,  however  small, — we  can  make 
the  Vs  as  near  together  as  we  please  if  only  we  do  not 
please  to  make  them  meet — between  them  stands  their 
common  limit,  2,  fringed  on  both  sides  with  a  row  of 
ratios  which  the  F's  in  their  race  towards  2  can  never  run 
through.  Now  consider  very  carefully  two  other  vari- 
ables, V  and  V'y  the  former  representing  any  one  of  the 
ratios  whose  square  is  less  than  2,  and  the  latter  any  one 
whose  square  is  greater  than  2.  Note  that  the  new  V's> 
like  the  old  ones,  can  come  indefinitely  near  together; 
observe  that  as  they  approach  each  other,  one  of  them 


288  MATHEMATICAL   PHILOSOPHY 

growing  continually  larger,  the  other  continually  smaller, 
though  they  can  never  meet,  yet  they,  like  the  old  F's 
come  to  differ  by  less  than  any  preassigned  positive 
amount,  however  small.  Undoubtedly  the  new  F's,  like 
the  old  ones,  seem  thus  to  close  in  upon  a  common  limit. 
Do  they  do  so  in  fact?  If  they  do,  what  is  the  limit? 
If  there  be  one,  it  must,  we  are  sure,  be  something  having 
a  square  and  having  2  for  the  square;  but  this  something, 
if  it  exist,  can  not  be  one  of  the  things  which  we  and 
our  race  have  hitherto  meant  by  number,  for,  by  hypoth- 
esis, the  only  numbers  we  know  in  our  present  stage  of 
evolution  are  cardinals,  integers  and  ordinary  fractions, 
and  none  of  these  has  2  for  its  square.  You  sense  vividly, 
I  trust,  the  painful  situation  into  which  our  limit  idea  has 
brought  us.  Do  you  know  how  we  will  behave  under  the 
circumstances?  How  we  will  try  to  escape?  By  what 
means  we  will  endeavor  to  reach  a  reconciliation?  We 
are  to  suppose  ourselves  to  be  dealing  with  the  difficulty 
as  the  mathematicians  have  dealt  with  it.  Accordingly, 
we  will  not  all  of  us  behave  in  the  same  way — some  of 
us  will  resort  to  one  means  of  extrication  and  some  to 
another.  {A)  Some  of  us  will  say:  V  and  V  have  not 
a  common  S  limit,  but  they  have  a  common  S"  limit  where 
Sr  is  a  sequence  of  things  we  have  not  yet  learned,  but 
must  learn,  to  recognize  and  handle;  this  common  limit, 
though  not  a  number  in  the  accepted  sense  of  the  term,  is 
something  we  must  regard  as  having  a  square  and  as 
having  2  for  its  square;  we  will  denote  the  thing  by  the 
symbol,  \/2,  and  call  it  a  number  of  a  new  kind — an 
irrational  number  to  distinguish  it  from  the  old  familiar 
ratios  to  be  henceforth  called  rational.  (B)  Others  of  us 
will  say:  V  and  V  have  no  common  limit  of  any  kind 
familiar  or  unfamiliar;   it  is,  however,  manifest  that  they 


MORE  ABOUT  LIMITS  289 

ought  to  have  one  for  the  sake  of  our  convenience,  and,  as 
none  exists,  we  will  create  one;  we  will  call  the  creature 
"  the  square  root  of  2,"  denote  it  by  the  symbol,  \/2y 
regard  it  as  a  number,  and  describe  it  as  irrational  to 
distinguish  it  from  the  old  sort  of  numbers, — the  ratios, — 
to  be  henceforth  described  as  rational.  (C)  Yet  others  of 
us,  not  so  numerous  but  harder-headed  and  more  critical, 
will  say:  it  is  evident  that  to  escape  decently  from  our 
predicament  we  must  somehow  enlarge  our  conception  of 
number;  not,  however,  by  asserting  that  V  and  V  have 
a  mysterious  sort  of  common  limit,  for  they  evidently 
have  no  common  limit;  nor  by  pretending  to  "  create  " 
one  for  them,  which  we  can  not  do;  but  by  discovering 
that  certain  existing  things,  not  hitherto  regarded  as 
numbers,  ought  to  be  so  regarded — a  discovery  that, 
briefly  sketched,  runs  as  follows:  we  reflect  that  the 
ranges  of  Vx  and  V  (we  could  equally  well  use  V2  and  V) 
are  classes  of  ratios  ordered  by  S;  we  observe  that  neither 
of  the  ranges  contains  a  maximum  term,  a  largest  ratio, 
though  one  of  them  (or  its  variable  Vi)  has  an  upper 
limit,  2,  and  the  other  has  no  upper  limit;  giving  the 
name  segment  to  such  ratio  ranges,  that  is,  to  such  of  them 
as  have  no  maximum,  we  see  that  a  segment  may  or  may 
not  have  an  upper  limit;  we  readily  see  that  segments 
have  certain  properties  (summability,  and  so  on)  very 
like  the  properties  of  what  we  have  been  calling  numbers; 
we  accordingly  and  naturally  agree  to  call  the  segments 
themselves  numbers;  they  are  a  new  kind  of  numbers — 
— not  ratios,  but  certain  classes  thereof;  we  call  the  new 
numbers  rational  if  the  segments  have  upper  limits  and 
irrational  if  they  have  not;  thus  the  segment  represented 
by  Vx  is  a  rational  number  while  that  represented  by  V 
is  irrational;    we  denote  the  former  by  2  because  the 


290  MATHEMATICAL   PHILOSOPHY 

upper  limit  of  J{  is  the  ratio  f ,  but  we  must  not  confound 
this  rational  2  (which  is  a  segment  of  ratios)  with  the 
cardinal  2  nor  with  the  integer  2,  nor  with  the  ratio  f 
(commonly  denoted  by  2)  nor  with  any  other  number  that 
blundering  custom  may  yet  denote  by  the  same  symbol; 
the  irrational  number  or  segment  represented  by  V  is 
denoted  by  \/2;    and  so  on   analogously  for  analogous 

cases;  to  each  ratio  7  will  correspond  a  rational  number  7, 
b  b 

so  that,  for  example,  to  the  ratio  f  will  correspond  the 
conceptually  distinct  rational  number  f ;  to  the  irrational 
numbers,  however,  no  ratios  will  thus  correspond;  the 
rationals  and  the  irrationals  taken  together  we  will  call* 
real  numbers;  these  may  be  arranged  in  the  order  of 
increasing  magnitude  by  a  sequence  S',  and,  if  we  then 
let  V\  range  be,  not  the  class  of  ratios  less  than  the  ratio  2, 
but  the  class  of  rationals  less  than  the  rational  2,  V  will 
indeed  have  2 — rational  2 — for  upper  limit;  and  so  at 
length  the  mystery  is  dispelled — what  fooled  us  before 
was  our  confounding  the  familiar  class  of  ratios  with  the 
then  unknown,  yet  vaguely  felt,  class  of  rationals,  corre- 
sponding to  but  logically  distinct  from  the  ratios. 

Well,  what  is  it  that  has  happened  here  in  our  racial 
history?  I  hope  you  see  that  what  has  happened  is  this: 
we  have  made  a  leap,  an  immense  forward  leap,  in  the 
course  of  mathematical  evolution;  we  have  made  a  great 
number-generalization;  we  have,  that  is,  extended  our  old 
familiar  well-established  concept  and  name  of  numbers 
so  as  to  make  it  include  and  cover  two  immense  new 
varieties,  namely,  the  rationals, — which  are  as  multitu- 
dinous as  the  infinite  host  of  our  old  traditional  ratios, — 
and  the  irrationals, — which  may  be  shown  to  be  infinitely 
more  in  multitude  than  all  the  old  numbers  taken  together. 


MORE  ABOUT  LIMITS  291 

Do  not  fail  to  observe  how  the  tremendous  generaliza- 
tion, so  copiously  enriching  our  human  world  of  mathe- 
matical ideas,  was  brought  about:  we  were  operating  in  a 
certain  domain,— the  domain  of  ratios;  we  were  there 
employing  the  notion  of  limit;  using  the  notion,  we  found 
ourselves  looking  for  a  limit  where  we  were  suddenly  and 
painfully  astonished  to  find  there  was  none;  we  were 
baffled,  we  wondered,  felt  a  need, — the  need  of  a  deeper 
view,  of  a  larger  vision,  of  a  more  embracing  conception 
to  extricate  us;  and  we  found  it — how?  By  means  of  the 
limit  idea;  that  which  got  us  into  the  difficulty  got  us 
out  of  it  and,  in  doing  so,  gave  us  a  larger  world. 

Did  the  limit  concept  compel  the  generalization?  No; 
such  generalization  is  never  compelled, — it  is  suggested, 
recommended,  stimulated,  even  urged,— but  not  com- 
pelled as  a  conclusion  from  premises, — generalization  al- 
ways involves  an  act  of  will, — a  choice  between  a  smaller, 
meaner  view  and  a  larger,  nobler  one;  and  in  the  present 
instance  it  was,  you  see,  the  notion  of  limit  that  gave 
man's  will  the  necessary  suggestion,  incitement  and 
guidance. 

There  is  another  aspect  of  the  matter  which  you  as 
philosophers  must  on  no  account  fail  to  notice  very 
carefully,  for  it  is  a  phenomenon  of  all  genuine  generaliza- 
tion. It  is  this:  the  world  of  the  real  numbers,  though 
itself  a  strictly  actual  world  once  it  is  found,  yet  is,  for 
any  possible  point  of  view  in  the  domain  of  ratios,  a 
strictly  ideal  world — ideal  in  the  just  sense  that,  though 
it  is  suggested  by  phenomena  in  the  domain  of  ratios,  it  is 
itself  wholly  outside  thereof  and  can  in  no  wise  be  attained 
by  pursuing  sequences,  however  endless,  within  the 
domain;  a  generally  neglected  fact  of  the  utmost  impor- 
tance,  not   only   in   discussing  the   spiritual  bearings  of 


292  MATHEMATICAL   PHILOSOPHY 

mathematics,  but  also  and  especially  in  understanding  the 
ways  of  spiritual  life — the  ways  of  truth  to  men. 

Not  to  wish  to  dwell  in  this  insight  long  enough  to 
make  it  our  own  would  show  us  unworthy — stupid  or 
perverse.  We  might  indeed  illustrate  it  in  many  ways  and 
in  many  connections.  We  might  show  in  detail  how  limit 
concepts  at  work  in  the  domain  of  real  numbers,  especially 
in  connection  with  equations  failing  to  have  roots  in  that 
domain  when  the  variable  coefficients  are  allowed  to 
approach  certain  limits,  make  us  keenly  aware  that  the 
domain  of  reals,  vast  as  it  is,  is  yet  too  meagre  for  our 
purposes,  and  how  we  are  thus  led  to  effect  another 
immense  number-generalization — that  one,  I  mean,  which 
gave  us  what  we  call  the  complex  numbers  (x+iy,  x  and  y 
being  reals,  and  i  being  the  so-called  imaginary  unit, 
■\/  —  i),  now  the  subject  of  a  stately  theory  having  wide 
application  in  physics  and  even  in  engineering;  we  might 
show  in  detail,  little  step  by  step,  how  limit  concepts  at 
work  in  geometry  have  availed  so  to  extend  or  generalize 
such  fundamental  notions  as  length,  area  and  volume — 
formerly  clear  and  well  defined  only  in  connection  with 
broken  lines,  or  polygons,  and  solids  bounded  by  planes — 
that  we  can  now  confidently  and  understandingly  use  the 
notions  in  connection  with  all  manner  of  curves  and 
curved  surfaces.  But  such  details  would  require  certainly 
more  time  and  perhaps  more  patience  than  we  now  have 
at  our  disposal. 

I  must,  however,  once  more  insist  upon  the  matter 
which  I  mentioned  a  moment  ago  and  which  I  have  em- 
phasized elsewhere.  The  matter  is  this:  a  limit-begotten 
generalization  always  originates  in  the  work  of  some  limit 
concept  operating  in  some  established  domain  (such  as  that 
of  our  ratios,  for  example)  wherein  the  concept   leads  us 


MORE  ABOUT  LIMITS  293 

into  the  presence  of  baffling  phenomena,  waking  our 
wonder,  giving  us  a  painful  sense  of  failing  to  see  some- 
thing we  ought  to  see,  a  sense  of  logical  suffocation,  of  be- 
ing hampered,  hemmed  in;  we  seek  emancipation  and  at 
length  achieve  it,  not  solely  by  purely  logical  means,  but 
partly  by  observation  (as  in  the  case  of  the  segments), 
partly  by  reasoning  and  partly  by  an  act  of  will — in  short, 
by  generalization;  this  deed  gives  us  a  new  domain  of 
thought — a  new  field  of  ideas  (as,  for  example,  the  domain 
of  real  numbers);  the  new  domain,  once  thus  established, 
is  as  actual  for  us  as  the  old  one;  with  reference,  how- 
ever, to  any  viewpoint  within  the  old  one,  the  new  domain 
is  and  forever  remains  a  sheer  ideal,  not  to  be  attained  by 
any  process  or  operation — however  oft  repeated,  swift  or 
prolonged — within  the  old  domain;  and  finally,  a  new 
domain  (as  that  of  the  real  numbers,  for  example)  may 
in  its  turn  become,  in  the  manner  indicated,  an  old  one 
in  relation  to  another  domain  (as,  for  example,  the  domain 
of  the  complex  numbers)  which,  though  itself  actual,  is, 
with  respect  to  the  former,  an  eternal  ideal. 

Mathematical  Limit  Processes  Viewed  as  Species  of 
Idealization. — In  nearing  the  close  of  this  second  long 
lecture  on  variables  and  limits  we  come  now  to  what  I 
most  desire  to  signalize  as  being  for  students  of  philosophy 
the  most  significant  aspect  of  the  whole  matter.  It  is  this: 
In  mathematics  the  great  role  of  what  we  there  call  limits 
and  limit  processes  is  in  kind  identical  with  the  momentous 
role  of  that  which  in  other  fields  of  interest  we  call  ideals 
and  idealization.  In  the  light  of  the  foregoing  discussion 
the  fact  is  evident,  and  it  shows  us  again  very  clearly 
what  we  have  repeatedly  seen  in  other  connections — that 
— far  from  being  detached  from  common  life  or  alien 
thereto, — mathematics  is   a   refined   model  or  prototype 


294  MATHEMATICAL   PHILOSOPHY 

of  that  which  in  life  is  most  precious  and,  strange  to  say, 
most  omnipresent,  too,  though  the  presence  be  often 
disguised.  With  sequences, — many  of  them  finite,  many 
of  them  potentially  infinite, — our  concrete  life  is  indeed 
replete:  sequences  of  potential  degrees  of  knowledge,  of 
potential  degrees  of  wisdom,  of  potential  degrees  of  skill, 
of  justice,  of  beauty,  of  righteousness,  of  authority,  of 
power,  of  freedom,  of  potential  degrees  of  innumerable 
forms  of  excellence  in  the  achievements  and  dreams  and 
aspirations  of  mankind.  I  hope  you  will  not  fail  to  see 
clearly  that  just  as  in  the  mathematical  prototype  of 
idealization, — in  the  theory,  that  is,  of  limits  and  limit 
processes, — so  here,  in  our  concrete  and  passionate  life, 
ideals  are  of  two  kinds:  namely,  ideals  which  we  pursue 
endlessly  from  degree  to  degree  of  excellence  of  a  given 
type,  as  a  variable  having  a  limit  endlessly  pursues  it 
without  attaining  it;  and  those  higher  ideals  which  are 
indeed  not  as  limits  of  endless  sequences  of  degrees  of 
excellence  of  a  given  type  (that  of  jazz  music,  for  example, 
or  that  of  a  Beethoven  sonata),  but  which  require  us  to 
rise  from  given  types  to  higher  types  by  a  species  of  ideali- 
zation corresponding  to  that  which,  in  the  model,  we  have 
called  limit-begotten  generalization.  It  is  thus  evident 
that  ideals  are  not  things  to  gush  over  or  to  sigh  and 
sentimentalize  about;  they  are  not  what  would  be  left  if 
that  which  is  hard  in  reality  were  taken  away;  ideals  are 
themselves  the  very  flint  of  reality,  beautiful,  no  doubt, 
and  precious,  without  which  there  would  be  neither  dig- 
nity nor  hope  nor  light;  but  their  aspect  is  not  sentimental 
and  soft;  it  is  hard,  cold,  intellectual,  logical,  austere. 
Idealization  consists  in  the  conception  or  the  intuition  of 
ideals  and  in  the  pursuit  of  them.  And  ideals,  I  have  said, 
are  of  two  kinds.     Let  us  make  the  distinction  clearer. 


MORE  ABOUT  LIMITS  295 

Every  sort  of  human  activity, — shoeing  horses,  abdominal 
surgery  or  painting  profiles, — admits  of  a  peculiar  type  of 
excellence.  No  sort  of  activity  can  escape  from  its  own 
type,  but  within  its  type  it  admits  of  indefinite  improve- 
ment. For  each  type  there  is  an  ideal, — a  dream  of  per- 
fection,— an  unattainable  limit  of  an  endless  sequence  of 
potential  ameliorations  within  the  type  and  on  its 
level.  The  dreams  of  such  unattainable  perfections  are 
as  countless  as  the  types  of  excellence  to  which  they 
respectively  belong  and  they  together  constitute  the 
familiar  world  of  our  human  ideals.  To  share  in  it, — 
to  feel  the  lure  of  perfection  in  one  or  more  types  of 
excellence,  however  lowly, — is  to  be  human;  not  to  feel 
it  is  to  be  sub-human.  But  this  common  kind  of  ideal- 
ization, though  it  is  very  important  and  very  precious, 
does  not  produce  the  great  events  in  the  life  of  mankind. 
These  are  produced  by  the  kind  of  idealization  that  corre- 
sponds to  what  we  have  called,  in  the  mathematical 
prototype,  limit-begotten  generalization, — a  kind  of  ideal- 
ization that  is  peculiar  to  creative  genius  and  that, 
not  content  to  pursue  ideals  within  established  types  of 
excellence,  creates  new  types  thereof  in  science,  in  art, 
in  philosophy,  in  letters,  in  ethics,  in  education,  in  social 
order,  in  all  the  fields  and  forms  of  the  spiritual  life  of  man. 
We  have  here,  you  see,  a  new  way — I  think  it  a  most 
fruitful  way — to  study  the  phenomena  of  spiritual  life, 
whether  our  own  or  that  of  mankind  in  general.  I 
leave  it  to  you  to  pursue  it  if  you  will,  for  that  is  what 
philosophy  is, — the  study  of  the  phenomena  of  the 
spiritual  life  man, — and  if  it  is  not  that,  it  is  nothing. 
In  relation  thereto  I  will  merely  say,  in  closing,  that  of 
the  two  kinds  of  ideals  and  idealization,  it  is  byjmeditating 
on  the  higher  kind,  in  the  light  of  the  mathematical  theory 


296  MATHEMATICAL   PHILOSOPHY 

of  limits,  that  I  have  been  led,  as  already  indicated,  to 
regard  the  great  Platonic  Absolutes  as  supernal  ideals, 
indicated  indeed  in  the  world  of  logic  but  there  indicated 
as  having  their  locus  above  it  so  that  they  appear  like 
downward-looking  aspects  of  an  over-world. 


LECTURE  XV 
Infinity 

MATHEMATICAL  INFINITY — ITS  DYNAMIC  AND  STATIC 
ASPECTS NEED  OF  HISTORY  OF  THE  IMPERIOUS  CON- 
CEPT  THE  ROLE  OF  INFINITY  IN  A  MIGHTY     POEM 

NO  INFINITY,  NO  SCIENCE. 

It  is  inconceivable  that  a  course  of  lectures  having 
the  aim  of  the  present  course  should  be  altogether  silent 
respecting  the  mathematical  concept  of  infinity.  For 
among  the  great  mathematical  concepts  that  are  acces- 
sible to  laymen  there  is  none  which  surpasses  this  one  in 
importance  or  in  power;  there  is  none  that  appeals  more 
strongly  to  the  imagination  of  such  as  are  qualified  to 
receive  it;  and  the  nearer  mile-posts  of  its  endless  avenue 
of  increasing  wonders  are  not  difficult  to  reach.  On  these 
accounts  the  temptation  to  devote  at  least  one  lecture  to 
an  elementary  exposition  of  the  idea  is  strong.  I  have 
decided,  however,  not  to  yield  to  it  for,  if  I  did  so,  I 
should  be  wasting  your  time;  I  should  be  only  adding  an- 
other one  to  already  numerous  expositions  of  the  kind, 
some  of  them  so  simple  and  clear  as  to  leave  no  excuse 
for  not  acquiring  a  fair  knowledge  of  the  matter  except 
the  melancholy  excuse  of  spiritual  inaptitude  therefor. 
In  the  Revue  de  Metaphysique  et  de  Morale,  for  example, 
the  elements  of  the  subject  are  handled,  often  admirably, 
in  a  variety  of  papers,  some  of  them  by  notable  philoso- 

297 


298  MATHEMATICAL    PHILOSOPHY 

phers  and  some  of  them  by  distinguished  mathematicians. 
One  of  the  best  expositions  I  have  seen  is  that  by  Profes- 
sor E.  V.  Huntington  in  his  The  Continuum  and  Other 
Types  of  Serial  Order  where,  moreover,  you  can  learn 
what  mathematicians  mean  by  the  highly  important  term 
"continuum," — the  Grand  Continuum  as  Sylvester  called 
it, — an  idea  with  which  for  lack  of  time  I  was  unable  to 
deal  in  the  two  preceding  lectures — and  where,  too,  you 
will  find  an  introduction  to  the  transfinite  numbers  of 
Georg  Cantor,  masterful  primate  among  all  who  have 
contributed  to  our  understanding  of  mathematical  infinity. 
For  another  excellent  account  I  may  refer  you  to  the 
already  mentioned  Fundamental  Concepts  of  Algebra  and 
Geometry  by  Professor  J.  W.  Young.  Clear  indication 
of  the  philosophic  significance  of  the  idea  in  question  is 
found  in  an  article  on  the  "Concept  of  the  Infinite"  by 
the  late  Professor  Royce  (Hibbert  Journal,  Vol.  I)  and 
in  the  Appendix  to  The  World  and  the  Individual,  by  the 
same  author.  Perhaps  no  one  else  has  treated  the  mat- 
ter with  so  much  deserved  emphasis  and  with  so  much 
freshness  and  facility  as  Bertrand  Russell  in  his  more 
popular  works.  For  some  indication  of  my  own  views 
respecting  the  bearings  of  the  concept  upon  certain  funda- 
mental questions  of  philosophy,  theology  and  religion,  I 
may  be  permitted  to  refer  you  to  Science  and  Religion; 
to  The  New  Infinite  and  the  Old  Theology;  and  to  the 
articles— "The  Walls  of  the  World,"  "The  Axiom  of  In- 
finity," and  "Mathematical  Emancipations" — contained  in 
The  Human  Worth  of  Rigorous  Thinking.  In  the  fore- 
going works,  you  will  find  an  ample  clue  to  the  extensive 
literature  of  the  subject,  both  that  which  is  more  popular 
and  that  which  is,  I  will  not  say  more  scientific,  but  more 
technical,    if   indeed   you    should,    fortunately,    desire    to 


INFINITY  299 

pursue  the  doctrine  in  its  elaborate  and  recondite  develop- 
ments. 

To  an  audience  of  philosophical  students  it  need  not 
be  said  that  some  notion  of  infinity  has  figured  conspicu- 
ously, often  fundamentally  and  dominantly,  throughout 
the  whole  historic  period  of  philosophy  and  speculation 
East  and  West.  It  may,  however,  be  said  to  such  an 
audience,  and  I  think  it  should  be  said,  that  a  critical  his- 
tory of  the  concept  of  infinity — or  rather  of  the  concepts 
thereof,  for  there  have  been  many  of  them,  for  the  most 
part  but  ill  defined — would  be  an  invaluable  contribution 
to  the  history  of  Thought — an  incomparably  more  im- 
portant contribution  than  the  philosophical  doctor  dis- 
sertations commonly  accepted.  There  can  hardly  be  a 
doubt,  I  believe,  that  the  mentioned  task  of  historical 
criticism  will  sometime  be  performed.  Why  should  it  not 
be  done  by  one  of  you?  You  are,  of  course,  aware  that 
the  doing  of  it  calls  for  an  extraordinary  kind  of  com- 
posite scholarly  preparation — linguistic,  historical,  philo- 
sophical, scientific,  and  especially  mathematical.  Our 
American  universities  have  long  been  amply  equipped 
with  adequate  machinery  for  the  giving  of  such  prepara- 
tion. Perhaps  one  of  you  will  demonstrate  that  they  have 
at  length  acquired  the  necessary  spirit  and  purpose  and 
atmosphere  and  temper. 

In  any  adequate  historico-critical  survey  of  the  role 
which  the  notion  of  infinity  has  played  in  our  human  think- 
ing, the  thought  of  many  thinkers,  widely  distributed  in 
time  and  in  space,  would  have  to  be  passed  in  review — 
analyzed,  understood,  and  appraised.  Among  the  ques- 
tions which  the  critic  would  have  to  ask  and  try  to  answer 
respecting  each  thinker  are  such  as  these:  What  did  he 
mean  by  infinite?     Did  he  employ  the  term  to  denote  a 


300  MATHEMATICAL   PHILOSOPHY 

definite  concept  or  at  best  a  vague  and  emotional  intui- 
tion? Was  his  thought  and  use  of  it  mystical,  or  logical 
and  analytical,  or  both?  Did  he  regard  his  infinite  as 
a  fact  or  as  an  hypothesis,  and  why?  Was  it  time?  An 
extension  in  time?  Space?  An  extension  in  space?  Was 
it  matter  or  mind  or  both?  Was  it  physical  or  spiritual? 
Concrete  or  abstract?  Did  he  define  it  and,  if  so,  in 
what  terms?  Or  did  he  take  it  as  a  primitive,  and,  if  so, 
did  he  do  it  consciously?  Did  he  think  of  it  as  magnitude 
or  as  multitude  or  as  both?  Had  he  but  one  infinite  or 
many  of  them?  If  many,  were  they  coordinate  or  hier- 
archical? If  the  latter,  was  the  hierarchy  crowned  or 
summitless?  Was  his  infinite  subordinate  in  his  thought 
or  central  and  dominant?  Did  he  employ  it  consistently 
or  confusedly?  Was  its  function  poetic  or  scientific  or 
both?  What  was  its  relation  to  the  modern  concept  of 
mathematical  infinity? 

It  has  seemed  to  me  that  I  could  best  serve  you  in  this 
hour  by  sketching  what  I  conceive  should  be  an  impor- 
tant chapter  in  such  a  critical  work.  The  sketch,  which 
will  be  very  imperfect,  is  offered,  not  as  a  model,  but  only 
as  a  concrete  suggestion.  I  have  selected  for  the  purpose 
the  philosophy  of  Lucretius,  in  which,  as  you  are  no  doubt 
aware,  the  notion — or  some  notion — of  infinity  is  very 
conspicuous.  The  question  is:  what  notion  and  what  is 
its  significance  there? 

It  will  facilitate  the  discussion  if  we  first  remind  our- 
selves of  the  meaning  of  mathematical  infinity  and,  in 
connection  therewith,  note  one  or  two  distinctions  and 
make  the  acquaintance  of  two  important  technical  terms 
— equivalence  of  classes,  and  denumer ability .  In  the  lec- 
ture on  the  nature  of  mathematical  transformation,  we 
met  the  notion  of  an  infinite  class  of  terms,  or  objects  of 


INFINITY  301 

thought;  we  there  saw,  for  example, — what  any  one  but 
a  fool  can  see, — that  we  can  set  up  a  one-to-one  corre- 
spondence between  the  integers  of  the  entire  class  of  in- 
tegers and  the  integers  (say  the  even  ones)  composing  a 
part,  or  sub-class,  of  the  entire  class,  by  the  simple  law  or 
device  of  making  i  correspond  to  2,  2  to  4,  3  to  6,  4  to 
8,  .  .  .  ,  n  to  2«,  and  so  on  endlessly.  Two  classes 
between  which  it  is  possible  to  set  up,  by  some  law  of 
transformation,  a  one-to-one  correspondence  are  said  to 
be  equivalent  classes;  and  a  class  that  is  equivalent  to  a 
part,  or  sub-class,  of  itself  is  called  an  infinite  class.  In- 
finite classes, — of  numbers,  of  points,  of  lines,  of  curves, 
of  surfaces,  of  propositions,  of  relations,  of  functions, 
and  so  on, — abound  on  every  hand;  theoretically,  and 
therefore  practically,  infinite  classes  are  more  important 
than  finite  ones,  even  though  the  spiritually  blind  are  un- 
able to  see  the  fact;  without  infinite  classes,  as  the  late 
Henri  Poincare  repeatedly  said,  there  could  be,  strictly 
speaking,  no  such  thing  as  science.  Scien^eJsJndfeedLthfi. 
study  of  infinity. 

""Tf  an  infinite  class  be  equivalent, — in  the  sense  de- 
fined,— to  the  class  of  positive  integers,  it  is  said  to  be 
denumerable — a  denumerably  infinite  class.  Many  in- 
finite classes  are  denumerable  which  have  not  the  appear- 
ance of  being  so.  A  striking  example  is  the  infinite  class 
of  our  ordinary  ratios,  or  fractions.  Between  any  two 
integers, — nay,  between  any  two  fractions,  however  near 
to  each  other  in  value, — there  are  infinitely  many  frac- 
tions— and  yet  the  entire  class  of  fractions  is  precisely 
equivalent  to  the  class  of  integers:  an  astonishing  fact 
readily  shown  as  follows.  In  the  preceding  lecture  we 
saw  that  the  fractions  can  be  arranged  in  a  row  by  means 
of  a  certain  rectangular  array  with  arrows.     To  see  that 


302  MATHEMATICAL   PHILOSOPHY 

the  equivalence  in  question  actually  exists,  you  have  now 
merely  to  observe  that  we  can  associate  the  first  frac- 
tion of  the  row  with  the  integer  I,  the  second  with  2,  and 
so  on,  thus  using  each  fraction  and  each  integer  once 
and  but  once.  Such  an  astonishing  result  makes  one 
wonder  whether  every  infinite  class  is  denumerable.  The 
answer  is,  No.  It  is  well  known  that  the  class  of  real 
numbers, — even  the  class  of  the  irrational  numbers,  even 
the  class  of  points  in  a  microscopically  short  line-segment, 
— is  non-denumerable.  Such  classes  are  infinite  but  they 
are  infinite  of  higher  order.  It  is  known  that  infinities 
rise  above  infinities  in  a  summitless  hierarchy.  At  present, 
the  denumerable  grade  and  that  of  the  real  numbers  are 
the  most  important.  In  time  to  come  such  may  not  be  the 
case ;  no  one  knows  enough  to  say. 

Here  we  must  make  a  distinction.  A  class  is  a  multi- 
tude— not  a  magnitude  such  as  length,  for  example,  or 
weight  or  area  or  volume  or  distance  or  the  like;  an  in- 
finite class  is  thus  an  infinite  multitude;  it  has  its  root  in 
the  question — how  many?  A  class  is  not  a  variable  in 
the  ordinary  sense  of  this  term, — it  is  a  fixed  thing, — a 
constant, — a  datum  given  once  for  all  in  the  world  of 
thought — in  logic  the  members  of  a  class  do  not  succeed 
each  other  in  time — they  coexist;  and  so  you  see  that  an 
infinite  class  is  a  static  infinity.  Long  before  this  concep- 
tion of  infinity  established  itself  in  mathematics  there  was, 
as  there  is  now,  another  conception  of  infinity, — of  a 
sort  of  dynamic  infinity, — namely,  the  conception  of  a 
changing  magnitude  or  function  capable  of  growing  to 
exceed  any  given  amount  denotable  by  any  integer  how- 
ever large.  The  idea  may  be  conveyed  as  follows:  let 
n   denote    any   given   positive    integer,    no    matter   how 


INFINITY  303 

large;  in  the  fraction  —f  let  x  be  a  variable  representing 

x 
a  real  number;  let  us  treat  x  as  an  infinitesimal — a  vari- 
able having  zero  for  its  limit;  as  x  grows  smaller  and 

smaller,  the  function  H  grows  larger  and  larger;  if  we 
x 

prescribe  any  finite  amount — an  amount,  that  is,  such  that 
we  can  denote  it  or  a  larger  one  by  a  positive  integer, 

then,  as  x  decreases  towards  zero,  —  will  come  to  exceed 

x 

the  prescribed  amount,  however  large;  we  express  this 

obvious  fact  by  saying  that,  for  x  approaching  zero,  — 

x 

approaches  infinity  or  becomes  infinite  or  is  an  infinite 
variable  or  function;  by  such  speech  mathematicians  do 
not  mean  that  there  is  a  definite  quantity  called  infinity 
(oo  )  and  that  the  ratio  becomes  equal  to  it  when  x  takes 
the  value  zero;  for  when  x  takes  this  value,  the  indicated 
division  becomes  meaningless  and  the  ratio  ceases  to 
exist;  what  the  speech  means — and  it  means  nothing  else, 
— is,  as  said,  that  for  x  decreasing  as  indicated,  the  ratio 
becomes  larger  than  any  prescribed  finite  amount.  Such 
is  the  conception  of  an  infinite  function  or  variable, — 
a  dynamic  infinity,  as  we  may  call  it  to  distinguish  it  from 
the  other, — the  static  infinity.  The  dynamic  type  has  its 
root  in  the  question — how  much?  It  is  obvious  that  the 
two  conceptions,  though  radically  distinct,  are  intimately 
related.  I  shall  leave  it  to  you  to  compare  them  deeply 
if  you  will. 

The  "new  infinity,"  as  it  is  sometimes  called,  means 
the  static  infinity.  It  was  introduced  into  mathematics 
something  more  than  a  half-century  ago  by  Bernhard 
Bolzano,   Richard  Dedekind  and  Georg  Cantor.     Long 


304  MATHEMATICAL   PHILOSOPHY 

before  that,  however,  it  was  grappled  and  wrestled  with 
by  two  geniuses  of  the  first  rank, — Galileo  (in  The  Two 
New  Sciences)  and  Pascal  (in  the  Pensees,  Havet's  edi- 
tion). 

Is  the  concept,  as  some  non-mathematicians  have  con- 
tended, a  mere  curiosity?  The  contention  springs  out  of 
the  unpardonable  academic  sin  of  stupidity. 

With  the  preliminaries  in  mind,  let  us  turn  to  the  De 
Rerum  Natura  of  Lucretius.1  This  work  of  an  Italian 
poet  I  have  already  mentioned,  in  the  lecture  on  the 
notion  of  group,  in  connection  with  what  I  have  there 
called  the  philosophy  of  the  cosmic  cycle,  or  cosmic  year. 
And  I  have  mentioned  it  as  one  of  the  greatest  works, 
not  of  a  Roman  as  such,  but  of  Man.  Memorable  on 
numerous  accounts  the  Romans  were.  For  the  construc- 
tion of  palaces,  temples,  roads,  aqueducts  and  other  public 
works — with  a  measureless  appalling  waste  of  material 
and  human  energy,  owing  to  pathetic  ignorance  of  science; 
for  inventions  in  the  art  of  war,  conquest  and  public 
murder;  for  elaborate,  sometimes  clever,  often  crude  and 
vulgar  imitation  of  Greek  letters,  eloquence,  and  art; 
for  a  manifold  development  of  an  imperious  jurispru- 
dence; for  the  theory  and  practice  of  empire  over  sub- 
jugated peoples;  for  the  unintentional  dissemination  of 
Hellenic  culture,  which  most  of  them  despised,  through- 
out vast  portions  of  the  world;  for  the  establishment, — 
by  conquest,  exploitation  and  robbery, — of  an  unrivalled 
luxury  and  sensual  magnificence  rotting  the  moral  fiber 
of  both  rulers  and  ruled:  on  these  and  similar  accounts, 
the  Romans  are  indeed  memorable   forever.     But  they 

*The  following  discussion  is  partly  embodied  in  my  article  "The 
Role  of  the  Concept  of  Infinity  in  the  YVork  of  Lucretius"  in  the  Bulletin 
of  the  American  Mathematical  Society,  April,  1918.  The  article  was  re- 
printed in  The  Classical  Weekly,  January  27,  1919. 


INFINITY  305 

are  equally  memorable  on  other  accounts — for  their  lack 
of  any  genuine  spirit  of  philosophic  enquiry,  for  their 
lack  of  reverence  for  human  beings  as  human,  for  their 
stupid  belief  that  the  things  of  wisdom  could  be  pur- 
chased, and  especially  for  their  brutal  lack  of  scientific 
curiosity,  scientific  imagination  and  scientific  achievement. 
Even  the  one  really  great  exception — De  Reriitn  Natura 
— is,  in  respect  of  its  content,  Greek  in  origin — it  is,  as 
you  know,  Epicurean;  it  is,  nevertheless,  the  "one  really 
great  exception,"  for  the  thought  of  the  Greek  thinker 
stirred  the  great  genius  of  the  Italian  poet  to  its  depths; 
Lucretius  understood  it  and  he  cast  it  in  immortal  form 
for  the  edification  of  all  posterity.  So  far,  however,  as 
Romans  were  concerned,  the  Lucretian  work  "was  still- 
born, into  a  suffocating  atmosphere  of  vile  wealth  and 
military  oppression.  The  true  figure  to  represent  the 
classical  Roman  attitude  to  science  is  not  Lucretius,  but 
that  Roman  soldier  who  hacked  Archimedes  to  death  at 
the  storming  of  Syracuse." 

Most  of  the  many  great  merits  of  the  work  of  Lucre- 
tius have  been  long,  though  not  generally  nor  even  widely, 
recognized.  One  of  its  recognized  merits  is,  as  I  have 
already  said,  its  superb  daring, — the  unsurpassed  mag- 
nificence of  its  enterprise,  which  was  nothing  less  than 
to  show  forth  a  method  for  explaining  all  phenomena 
(whether  mental  or  not)  without  having  to  resort  to  any 
hypothesis  of  divine  intervention;  another  of  its  merits, 
— a  very  striking  one, — is  its  probably  unequalled  union 
of  literary  excellence  with  scientific  spirit  and  aim;  still 
another — which  includes  many,  being  a  highly  composite 
merit — is  its  confident  and  often  acutely  argued  presenta- 
tion, sometimes  in  detail  and  sometimes  in  clear  outline 
only,  of  ideas  and  doctrines  whose  just  recognition  had  to 


306  MATHEMATICAL   PHILOSOPHY 

await  the  coming  of  modern  science.  I  refer  to  such 
scientific  concepts  and  dogmas  as :  natural  law — the  atomic 
constitution  of  matter — conservation  of  mass — conserva- 
tion of  energy — organic  evolution — spontaneous  or  chance 
origination  and  variation  of  organic  life  forms — struggle 
for  existence  in  partly  friendly  and  partly  hostile  environ- 
ment— survival  of  the  fit  (the  well  adapted)  and  de- 
struction of  the  ill  adapted — and  sensation  as  the  ulti- 
mate basis  of  knowledge  and  as  the  ultimate  test  of  reality 
— not  to  mention  other  equally  brilliant  anticipations  of 
"modern"  scientific  thought. 

In  extant  appreciations  of  the  work  of  Lucretius  his 
employment  of  the  notion  of  infinity  is  indeed  commonly 
indicated  but  it  is  indicated  only  more  or  less  incidentally, 
without  due  signal  of  that  notion's  role  in  the  poet's 
thought.  For  example,  in  Masson's  large  and,  in  many 
ways,  excellent  volume, — Lucretius,  Epicurean  and  Poet, 
— the  term  infinite  has  only  a  subordinate  place  in  the 
index  of  important  terms;  in  the  very  extensive  Notes  to 
Munro's  famous  translation  the  term  receives  but  scant 
attention;  and  it  receives  even  less  in  the  Notes  found 
in  Cyril  Bailey's  recent  and  deservedly  much  praised 
English  translation  of  the  poem.  What  is  missed  in 
such  appreciations  and  commentaries  and  what  I  wish 
especially  to  signalize  here  is  the  fact  that  the  concept  of 
infinity, — of  infinite  multitude  and  infinite  magnitude, — 
is  not  merely  one  among  the  many  ideas  employed  by 
Lucretius  but  is  indeed  the  dominant  idea  in  his  system 
of  thought.  A  critical  examination  of  the  work  as  a  sci- 
entific structure  can  hardly  fail  to  discover  that  in  the 
author's  judgment  the  concept  of  infinity  was  not  only 
the  most  powerful  of  his  logical  instruments  but  also — 
which  is  quite  another  matter — the  one  most  obviously 


INFINITY  307 

indispensable  to  the  prosperity  of  his  great  undertaking. 
This  is  not  the  place  to  give  a  detailed  account  of 
the  Lucretian  principles  and  procedure.     For  our  present 
purpose  it  is  sufficient  to  point  out  that  among  the  propo- 
sitions of  De  Rerum  Natura  there  are  three  major  ones 
and  that  these  owe  their  efficacy  and  indeed  their  control 
of  the  entire  discourse  to  the  fact  of  their  postulating 
the  existence  of  infinite  multitude  and  infinite  magnitude. 
"Postulating"  I  have  said,  although  Lucretius  regarded 
the  propositions,  not  as  mere  hypotheses  or  assumptions, 
but  as  indubitable  certitudes.    What  are  these  three  basic 
propositions?    They  are:  that  the  universe  of  space  is  a 
region  or  room  of  infinite  capacity — infinite  extent;  that 
time  is  an  infinite  duration  composed  of  a  beginningless 
infinite  past  and  an  endless  infinite  future;  and  that  the 
universe's  matter  consists  of  an  infinite  multitude  of  ab- 
solutely solid  (non-porous)  and  non-decomposable  atoms 
■ — "seeds  of  things" — always  moving  hither  and  thither 
in   an  infinite  variety  of  ways   and  ever  so   distributed 
throughout  infinite  space  that  of  all  spheres  none  but  such 
as  are  microscopically  minute  could  at  any  instant  fail 
to  enclose  one  or  more  of  the  "seeds."     Without  these 
infinitudes,  explanation  of  the  phenomena  of  the  world 
was,  in  the  poet's  belief,  impossible;  with  them,  supple- 
mented by  certain  other  principles,  such  explanation  was 
possible.     In  the  view  of  Lucretius  cosmic  history  was  an 
eternal   (infinite)   drama  enacted  by  an  infinitude  of  un- 
originated  and  indestructible  "seeds,"  atoms,  or  elements 
operating  upon  an  infinite  stage.     The  drama  was  not  to 
be  understood  except  by  help  of  the  concept  of  infinity; 
and  so  De  Rerum  Natura  may  be  not  unjustly  said  to  be 
a  kind  of  poetic  celebration  of  what  the  poet  deemed  to 
be  the  scientific  efficacy  of  that  concept. 


308  MATHEMATICAL   PHILOSOPHY 

What  did  Lucretius  mean  by  infinity?  What  did  he 
mean  by  an  infinite  multitude  and  by  an  infinite  magni- 
tude? No  formal  definition  of  any  of  these  terms  is  to 
be  found  in  his  work.  It  is  perfectly  clear,  however,  that, 
if  one  had  asked  him  whether  an  infinite  multitude  of 
elements  was  such  that  it  could  not  be  exhausted  by  re- 
moving from  it  one  element  at  a  time,  he  would  have 
answered  in  the  affirmative;  and  that,  if  one  had  asked 
him  whether  the  elements  of  an  infinite  multitude  could 
be  thought  of  as  arranged,  like  beads  on  a  string,  in  an 
endless  succession  of  elements,  he  would  have  again  an- 
swered affirmatively.  In  short,  an  infinite  multitude  sig- 
nified for  Lucretius  what  mathematicians  now  describe 
as  a  denumerably  infinite  multitude  or  class.  In  his  work 
there  is  no  hint  or  suggestion  that  he  had  any  conception 
or  any  inkling  of  any  higher  order  of  infinity.  It  is  highly 
probable  or  indeed  quite  certain  that,  owing  to  his  lack 
of  mathematical  discipline,  such  a  conception,  had  it  been 
suggested,  would  have  seemed  to  him  unintelligible  or 
absurd. 

It  is  in  itself  noteworthy,  and  if  one  is  really  to  under- 
stand Lucretius  it  is  essential  to  note,  that,  with  the  pos- 
sible exception  of  time,  the  fundamental  Lucretian  infini- 
ties were  not  mere  variables  capable  of  increase  beyond 
any  prescribed  finite  amount — they  were  not,  that  is,  what 
we  have  called  dynamic  infinities;  on  the  contrary,  they 
were,  like  the  infinites  of  Cantor,  constant  or  static  af- 
fairs; but,  unlike  the  Cantor  infinites,  those  of  Lucretius 
were  composed  of  actual  concrete  things  and  not  of  ab- 
stract ones  like  points,  for  example,  or  pure  numbers; 
thus  the  Lucretian  infinitude  of  atoms,  for  example,  was 
an  infinitude  of  material  particles  (taking  up  room)  and 
they  all  existed  at  once. 


INFINITY  309 

Let  us  recall  the  current  definition  of  an  infinite  class : 
An  infinite  class  is  a  class  having  a  sub-class,  or  part, 
equivalent  to  the  whole — equivalent,  that  is,  in  the  sense 
that  a  one-to-one  correspondence  can  be  set  up  between 
the  elements  of  the  part  and  those  of  the  whole.  This 
definition  of  infinity  was  not  given  by  the  poet  for,  as 
we  have  seen,  he  gave  no  formal  definition  of  it  at  all. 
We  may  ask,  however,  whether  Lucretius  was  aware  of 
the  fact  that  an  infinite  multitude,  as  conceived  by  him, 
contained  parts,  or  sub-multitudes,  equivalent,  as  we  now 
say,  to  the  whole.  The  answer  is,  yes:  not  only  was  he 
aware  of  it  but  he  repeatedy  employed  this  characteris- 
tic property  of  infinite  multitudes  correctly  and  effectively. 
This  rather  astonishing  fact  is  sufficiently  interesting  to 
justify  citation  of  one  or  two  passages  supporting  my  as- 
sertation  of  it.  If  we  bear  in  mind  that  one  of  the  funda- 
mental Lucretian  infinites  was  the  succession  of  time  units 
(days,  say,  or  generations  or  other  finite  stretches)  begin- 
ning at  any  given  instant  and  together  composing  what 
is  called  the  future,  the  following  famous  passage  makes 
it  perfectly  clear  that,  according  to  its  author,  the  re- 
moval of  any  finite  multitude  of  elements  from  an  in- 
finite multitude  of  them  leaves  a  remainder — a  part — 
exactly  equal  (or  equivalent,  as  we  say)  to  the  whole: 

Nor  by  prolonging  life  do  we  take  one  tittle  from 
the  time  past  in  death  nor  can  we  fret  anything  away, 
whereby  we  may  haply  be  a  less  long  time  in  the  con- 
dition of  the  dead.  Therefore,  you  may  complete  as 
many  generations  as  you  please;  none  the  less,  how- 
ever, will  that  everlasting  death  await  you;  and  for  no 
less  long  a  time  will  he  be  no  more  in  being,  who  begin- 
ning with  today  has  ended  his  life,  than  the  man  who 
has  died  many  months  and  years  ago.1 

1  Munro's  translation,  4th  ed.(  p.  83. 


310  MATHEMATICAL    PHILOSOPHY 

Lucretius,  as  already  said,  postulated  the  existence  of 
an  infinitude  of  atoms.  These  "seeds  of  things" — by 
whose  clashings  together  and  interlockings  with  one  an- 
other all  things  (including  souls)  were  produced,  to  be 
sooner  or  later  again  resolved  into  their  elements  by 
ceaseless  hammering  of  atomic  storms — these  "seeds," 
the  ultimate  constituents  of  all  the  world  (including 
minds),  were  not  all  of  them  identical  in  shape  nor  in 
size,  though  all  of  them  were  too  minute  to  be  seen  singly 
or  to  be  thus  apprehended  by  any  other  sense;  in  respect 
to  shape  and  size  the  atoms  presented  a  number  of  va- 
rieties but  only  a  finite  number.  The  atoms  of  each 
variety,  it  was  held,  constituted  an  infinite  multitude ;  and 
so  there  was  some  finite  number  of  infinite  classes  of 
atoms.  The  physical  functions  of  the  atoms  of  one  class 
were,  in  virtue  of  their  size  and  shape,  different  from  the 
functions  of  the  atoms  of  any  other  class.  In  respect, 
however,  of  multiplicity,  these  infinite  classes  were  equiv- 
alent— they  were  each  of  them  denumerable — and  each 
of  the  classes  was  equivalent  to  the  class  which  we  today 
should  call  their  logical  sum, — to  the  class,  that  is,  of 
all  the  atoms  in  the  universe. 

It  is  sufficiently  evident  that  the  poet's  conception  of 
infinite  multitude  was  identical  with  that  now  employed 
by  mathematicians.  If  you  will  carefully  scrutinize  the 
poem,  you  will  discover  that  the  same  may  be  said  of  the 
author's  conception  of  infinite  magnitude.  Formal  defini- 
tion of  the  notion  is  not  present.  We  are  told,  however, 
that  all  the  atoms  are,  in  respect  of  size,  between  a  finite 
upper  bound  and  a  finite  lower  bound,  and  this  notion 
of  lower  bound  is  of  critical  importance — what  the  lower 
bound  is  we  are  not  told  but  we  are  told  that  there  is 
such  a  bound  and  that  it  is  finite   (not  zero), — in  other 


INFINITY  311 

words,  atomic  size  is  not  infinitesimal, — it  is  a  variable 
but  not  one  having  zero,  or  null  size,  for  limit;  we  are 
told,  rightly,  that  the  sum  of  any  finite  number  of  atoms 
is  finite;  we  are  told  that  the  sum  of  all  the  atoms  of  a 
given  atomic  form  is  infinite  and  that,  therefore,  their 
number  must  be  infinite.  It  is  thus  evident  that  the  Lu- 
cretian  conception  of  an  infinite  magnitude  was  that  of 
a  magnitude  exceeding  the  sum  of  any  finite  number  of 
finite  quantities  none  of  which  surpasses,  in  respect  of 
parvitude,  a  finite  size. 

It  is  important  to  bear  in  mind  that  formation  of 
ideas  or  possession  of  them  is  one  thing,  and  that  logically 
correct  handling  of  them  in  argumentation  is  quite  an- 
other. The  difference  is  that  between  conception  and 
ratiocination.  In  his  use  of  the  ideas  in  question  Lucre- 
tius was  frequently  right  and  frequently  wrong.  You 
would  find  it  a  very  edifying  discipline  to  determine  all 
the  instances  of  both  kinds.  Of  right  use  some  examples 
have  already  been  given  and  it  would  be  easy  to  cite 
others.  Let  us  now  consider  an  instance  of  erroneous 
use.  A  remarkable  example  is  found  in  the  following 
passage  (as  correctly  translated  by  Munro,  page  15)  — 
a  passage  of  exceeding  interest  apart  from  the  error  in 
question: 

Again  unless  there  shall  be  a  least,  the  very  small- 
est bodies  will  consist  of  infinite  parts,  inasmuch  as 
half  of  a  half  will  always  have  a  half  and  nothing  will 
set  bounds  to  the  division.  Therefore  between  the 
sum  of  things  and  the  least  of  things  what  difference 
will  there  be?  There  will  be  no  distinction  at  all;  for 
however  absolutely  infinite  soever  the  whole  sum  is, 
yet  the  things  which  are  smallest  will  equally  consist 
of  infinite  parts. 


312  MATHEMATICAL   PHILOSOPHY 

The  significance  of  the  passage  and  the  erroneous  use 
it  makes  of  the  concept  of  infinity  will  be  clearer  to  us 
if  we  observe  that  the  passage  is  a  portion  of  an  argument 
by  which  Lucretius  endeavors  to  prove  that  a  finite  por- 
tion of  matter  is  not  indefinitely  or  limitlessly  divisible. 
He  assumed,  as  we  have  seen,  that  matter  is  composed 
of  invisibly  small,  absolutely  solid  particles  called  atoms, 
or  "seeds  of  things,"  these  atoms  being,  by  hypothesis, 
the  smallest  particles  capable  of  existing  spatially  sepa- 
rate from  one  another.  He  conceived  an  atom,  however, 
to  be  composed  of  parts,  which  were,  of  course,  not  sep- 
arable spatially  from  the  atom.  His  contention  was  that 
among  the  parts  of  an  atom  there  was  a  least  part — a 
part,  that  is,  such  that  none  of  the  parts  was  smaller. 
The  foregoing  quotation  is,  as  I  have  said,  a  part  of  the 
poet's  argument  in  behalf  of  this  contention.  Para- 
phrased in  modern  terms  this  portion  of  the  argument 
would  run  about  as  follows:  "If  among  the  parts  com- 
posing an  atom  and  being  such  that  no  two  of  them  have 
points  in  common  (save  points  of  a  common  surface) 
there  be  no  least  part,  then  the  atom  consists  of  an  in- 
finite number  of  non-interpenetrating  parts;  the  infinite 
multitude  of  atoms  in  the  universe  and  the  infinite  mul- 
titude of  parts  of  one  atom  are,  as  multitudes,  equivalent 
(in  the  sense  of  one-to-one  correspondence  between  the 
atoms  in  the  former  multitude  and  the  atom-parts  in  the 
latter)  ;  the  sum  of  the  elements  (atoms)  of  the  multitude 
of  the  atoms  is  an  infinite  magnitude,  the  total  quantity 
of  the  universe's  matter;  so,  too,  the  sum  of  the  elements 
(atom-parts)  of  the  infinite  multitude  of  parts  of  one 
atom  is  an  infinite  magnitude;  but  this  latter  sum  is  the 
atom  itself;  hence,  if  there  be  no  least  part  among  the 
parts  of  an  atom,  an  atom  is  an  infinite  magnitude,  and 


INFINITY  313 

as  such  is  no  less  than  the  sum  of  all  matter."  The  error 
to  which  I  desire  to  invite  your  attention, — an  error  in 
the  poet's  use  of  the  concept  of  infinity, — is  his  assertion 
italicised  in  the  foregoing  paraphrase.  The  error  is  not 
due  to  wrong  conception  of  infinity,  whether  of  multitude 
or  of  magnitude;  it  is  due  solely  to  the  tacit  assumption 
that  the  sum  of  the  elements  of  any  infinite  multitude  of 
elements  is  infinite, — an  assumption  which,  as  you  are 
aware,  is  false,  for,  for  example,  the  sum  of  the  elements 
of  the  infinite  multitude  of  elements  (h  h  h  •  •  •)  ls> 
as  you  learned   in  high  school,   not  infinite  but   is    i  — 

in  other  words,  the  limit  of  the  sum,  I s,  of  the  first 

/•     i  •         i   ■    I   ■     i   i  I     i 

n   terms   or   the   series,  — I — ~+-^-r  .  .  .  ,  -.+  •  •  • 

2        22        23  2 

is  i.  Such  a  series, — any  series  such  that  the  sum  of 
the  first  n  terms  has  a  finite  limit  for  n  increasing  limit- 
lessly, — is  said  to  be  convergent.  An  obvious  moral  is 
that  a  little  knowledge  of  the  convergence  of  series 
would  greatly  improve  the  philosophy  of  poets  and  the 
science  of  philosophers. 

It  is  astonishing  that  the  mentioned  fallacy  occurs,  as 
it  does,  in  immediate  conscious  connection  with  a  line 
seeming  (to  us)  to  refute  it:  the  half  of  the  half  will 
alzvays  have  a  half  and  nothing  will  set  bounds  to  the 
division.  What  is  the  explanation?  It  is  not  to  be  found 
in  any  supposition  of  stupidity  or  of  momentary  nodding. 
It  is  doubtless  to  be  found  in  the  author's  purpose  and 
point  of  view.  He  was  here  exclusively  concerned  with 
natural  phenomena,  with  what  he  deemed  to  be  existing 
entities — with  bodies  (and  parts  thereof)  occupying 
space,  actually  filling  what  would  else  have  been  absolute 
emptiness  or  void.     And  so,  if  you  had  tried  to  refute 


314  MATHEMATICAL   PHILOSOPHY 

him  by  means  of  such  a  series  as  i,  i,  i,  .  .  .  ,  which 
his  own  words  indeed  suggest,  he  would  probably  have 
said  in  effect:  "Composed  of  man-made  symbols  like 
words,  your  series  is  not  and  never  can  be  endless;  to 
speak  of  the  sum  of  a  non-existing  endless  series  is  mean- 
ingless; moreover,  even  if  we  supposed  the  series  to  be 
endless,  to  be  summable  and  to  have  i  for  its  sum,  this 
would  be  neither  finite  nor  infinite,  for  it  would  not  be 
a  magnitude,  inasmuch  as  the  summands  are  themselves 
not  magnitudes  but  are  merely  empty  abstract  symbols; 
if  i  be  said  to  be  a  magnitude,  in  the  sense  of  representing 
a  magnitude,  then  indeed,  if  magnitude  I  be  composed 
of  two  equal  magnitudes,  I  grant  that  |  will  be  a  mag- 
nitude in  the  same  sense  (of  representing  one)  ;  if  all  the 
symbols  be  magnitudes  in  that  same  sense,  the  summation 
of  the  series  of  abstract  symbols  may  be  said  to  be  the 
summation  of  an  endless  (infinite)  series  of  magnitudes; 
but  otherwise,  not;  and  now  what  I  have  contended  in 
my  poem  is  that,  if  your  magnitude  I  be  finite,  not  more 
than  a  finite  number  of  the  symbols  in  the  series  can  be 
magnitudes,  and  this  contention,  denying  the  endless  di- 
visibility of  finite  magnitude, — especially  denying  that  an 
atom  has  an  infinitude  of  parts, — is  based  on  physical 
considerations — on  grounds  other  than  that  advanced  in 
the  passage  you  have  quoted  from  my  argument."  If 
Lucretius  thus  replied  to  you,  what  suitable  rejoinder,  if 
any,  could  you  make? 

I  shall  not  attempt  to  recount  here,  much  less  to  esti- 
mate, those  other  grounds.  It  must,  however,  be  said,  in 
passing,  that  one  of  them  is,  in  point  of  kind,  almost  per- 
fectly represented  by  the  following  words  of  Clerk  Max- 
well  (  Theory  of  Heat,  p.  285)  : 


INFINITY  315 

What  we  assert  is  that  after  we  have  divided  a  body 
into  a  certain  finite  number  of  constituent  parts  called 
molecules,  then  any  further  division  of  these  molecules 
will  deprive  them  of  the  properties  which  give  rise  to 
the  phenomena  observed  in  the  substance. 

The  traditional  form  of  the  thesis  tacitly  invoked  by 
Lucretius  to  fortify  his  "other"  grounds  for  holding  that 
among  the  parts  of  an  atom  there  is  a  least  part,  is  ex- 
ceedingly vague :  all  infinites  are  equal.  Its  vagueness 
helps  to  account  for  its  ages-long  and  world-wide  vogue. 
Thus  Kanadi,  an  old  Hindu  author,  employs  the  thesis 
to  prove  that,  if  every  body  be  infinitely  divisible,  there 
can  be  "no  difference  of  magnitude  between  a  mustard 
seed  and  a  mountain"  (Daubeny's  Introduction  to  Atomic 
Theory,  p.  5).  In  this  connection,  anyone,  philosopher 
or  mathematician,  if  he  be  at  all  interested  in  the  history 
of  the  idea  of  infinity,  will  be  glad  to  have  his  attention 
called  to  a  little-known  letter  of  Newton  dealing  with  the 
idea.  The  letter,  which  is  addressed  to  Richard  Bentley 
{Works  of,  Vol.  Ill,  p.  207),  is  interesting  on  several 
accounts:  it  points  out  the  vagueness  and  falseness  of  the 
above-mentioned  thesis,  which  Dr.  Bentley  had  assumed 
to  be  true;  it  itself  repeatedly  employs  the  term  "infinite" 
in  a  sense  not  less  vague  and  indeterminate;  and  it  virtu- 
ally asserts  that,  if  two  infinite  magnitudes  be  equal,  the 
addition  of  any  finite  magnitude  to  either  of  them  will 
destroy  the  equality — a  proposition  which  we  now  know 
to  be  false. 

I  have  said  that  a  thorough-going  Critical  History 
of  the  Concept  of  Infinity  would  be  a  highly  valuable  con- 
tribution to  our  knowledge  and  understanding  of  human- 
kind. The  account  I  have  now  given  of  the  role  of  in- 
finity in  the  work  of  Lucretius  is  submitted,  let  me  say 


316  MATHEMATICAL  PHILOSOPHY 

again, — not  as  a  model,  for  it  is  too  imperfect  for  that, 
— but  as  only  a  concrete  suggestion  of  what  one  chapter 
of  such  a  history  might  contain.  In  closing  the  lecture, 
I  desire  to  guard  against  the  danger  of  leaving  a  false 
impression.  The  mere  correctness  of  the  Lucretian  con- 
cept of  infinity  does  not  of  itself  account  for  the  signifi- 
cance and  power  of  the  author's  work.  The  secret  lies 
in  the  fact  that  the  imagination  of  a  great  thinker  and 
poet  was  so  stimulated  by  the  concept  as  to  cause  him  to 
express  in  immortal  form  a  body  of  ideas  which  he  had 
acquired  from  an  elder  and  alien  world  and  which  after 
the  long  lapse  of  centuries  are  found  to  be  among  the 
most  fruitful  scientific  ideas  of  our  time. 


LECTURE  XVI 
Hyperspaces 

MEANING     OF     DIMENSIONALITY SPACES     OF     FOUR     OR 

MORE  DIMENSIONS THE  MODE  OF  THEIR  EXISTENCE 

DISTINCTION   OF   IMAGINATION   AND   CONCEPTION 

LOGICAL  EXISTENCE  AND  SENSUOUS  EXISTENCE 

OPEN   AVENUES   TO   UNIMAGINABLE   WORLDS. 

It  is  the  aim  of  this  lecture  to  explain  in  simple  ways 
what  mathematicians  mean  by  Hyperspace  and  to  convey 
some  sense  of  the  scientific  and  philosophic  importance  of 
the  concept  which  the  term  denotes.  Let  us  understand 
in  the  first  place  that  the  terms, — "hyperspace,"  "multi- 
dimensional space,"  "space  of  n  dimensions,"  "w-dimen- 
sional  space,"  and  some  other  readily  recognized  variants 
upon  them, — are  but  different  names  for  one  and  the 
same  idea;  they  are  employed  interchangeably  as  equiva- 
lents; and  the  like  may  be  said  of  the  terms, — "geometry 
of  hyperspace,"  "multi-dimensional  geometry,"  "geome- 
try of  n  dimensions,"  and  their  variants, — each  of  which 
simply  denotes  the  geometry,  or  the  science,  of  a  space 
having  more  than  three  dimensions. 

The  concept  of  hyperspace,  though  it  is  a  modern 
notion,  is  not  strictly  new, — it  goes  back  three  or  four 
generations  and  is  now,  among  enlightened  mathemati- 
cians, as  classic  and  orthodox  as  the  ordinary  multiplica- 
tion table.     Though  only  a  short  while  ago  it  was  re- 

317 


318  MATHEMATICAL   PHILOSOPHY 

garded  by  mathematicians  of  the  conservative  and  reac- 
tionary type  with  a  good  deal  of  suspicion  as  being,  if  not 
crazy,  at  least  a  bit  queer,  over-romantic,  and  unsound, 
it  is  now  constantly  employed  as  a  great  convenience  by 
mathematicians  everywhere  and  even  by  physicists  (say 
in  the  kinetic  theory  of  gases)  quite  without  apology. 
The  literature  of  the  subject  is  large  and  growing.  In 
Sommerville's  Bibliography  of  Non-Euclidian  Geometry, 
Including  the  Theory  of  Parallels,  the  Foundations  of 
Geometry,  and  Space  of  n  Dimensions  (1911)  there  are 
listed  1832  references  on  n  dimensions. 

The  concept  has  not,  indeed,  so  great  scientific  and 
spiritual  dignity  as  some  others, — as  that  of  function,  for 
example,  or  relation  or  transformation  or  group  or  in- 
variance  or  infinity  or  limit, — yet  it  is  a  very  grave  notion, 
and  it  has,  moreover,  a  certain  double  distinction :  it  is, 
I  mean,  one  of  the  few  among  the  important  concepts  in 
modern  mathematics  that  philosophers  have  seriously 
grappled  with  and  one  of  the  still  fewer  that  have  piqued 
the  curiosity  of  the  educated  public.  The  results  of  such 
popular  curiosity  are  themselves  a  little  curious.  Not 
long  ago,  for  example,  I  heard  and  read  an  address  on 
hyperspace  which  a  professional  astronomer  had  ventured 
to  make  before  an  audience  of  university  students.  It 
was  not  a  happy  performance;  not  only  did  the  speaker 
confound  the  idea  of  ^-dimensional  space  with  that  of 
non-Euclidean  space,  but  he  made  it  pathetically  evident 
that  he  had  grasped  neither  the  one  idea  nor  the  other, 
nor  did  anyone  in  the  evidently  interested  audience  ap- 
pear to  observe  the  fact.  I  have  cited  this  instance  be- 
cause, if  a  reputable  astronomer  can  err  so  egregiously 
in  a  matter  that  is  not  remote  from  the  field  of  his  special 
studies,  we  ought  not  perhaps  to  be  astonished  at  the 


HYPERSPACES  S19 

meagreness  of  the  educated  layman's  understanding  of  it. 
And  yet  the  fact  is  astonishing.  For  interest  in  the  con- 
cept of  hyperspace  and  especially  in  what  is  naively  called 
the  idea  of  "the  fourth  dimension"  is,  as  you  know, 
widespread  among  educated  laymen;  the  concept  itself, 
as  we  are  going  to  see,  is  not  a  very  difficult  one;  and  fair 
accounts  of  it  have  been  given  from  time  to  time  in  popu- 
lar and  semi-popular  magazines  and  books.  Neverthe- 
less, understanding  of  the  matter,  outside  the  circle  of 
professional  mathematicians,  is  exceedingly  rare.  What 
is  the  explanation?  What  has  been  the  trouble?  No 
doubt  part  of  it  is  that  competent  mathematicians  have, 
in  general,  been  unwilling,  sometimes  haughtily  unwilling, 
to  explain  their  ideas  in  popular  terms  lest  they  should 
seem  to  be  thus  seeking  the  applause  of  the  gallery, — not 
aware  of  the  fact  that  such  haughtiness  is  itself  one  of 
the  most  effective  means  of  impressing  the  gallery  with- 
out enlightening  it,  winning  its  applause  of  what  it  is  per- 
mitted to  believe  is  a  kind  of  mysterious  intelligence  so 
high  and  mighty  as  to  be  inaccessible  to  all  mortals  save 
the  few  who  are  endowed  with  mathematical  genius;  no 
doubt  another  part  of  the  trouble  has  been  that,  though 
the  concept  of  hyperspace  has  indeed  aroused  wide  curi- 
osity, it  has  not  been  pursued  diligently  in  our  industrial 
generation  as  it  would  have  been  had  it  seemed  to  have 
practical  or  bread-winning  value, — if,  in  other  words, 
instead  of  being  only  a  form  of  spiritual  wealth,  it  had 
carried  the  promise  of  material  wealth.  These  consider- 
ations, however,  do  not,  I  believe,  explain  the  matter 
fully.  The  main  trouble  has  been  that,  though  the  idea 
in  question  is  not  very  difficult  to  acquire,  yet  the  acquisi- 
tion of  it  does  demand  some  patient  meditation,  some 
precision  of  thought,  and  the  exercise  of  a  little  genuine 


320  MATHEMATICAL   PHILOSOPHY 

wit,  and  this  is  a  price  that  the  vast  majority  of  "edu- 
cated" laymen  are  unwilling  to  pay;  their  interest  in  scien- 
tific ideas  is  neither  steady  nor  deep;  the  ideas  they  acquire 
are  such  as  can  be  taken,  so  to  speak,  on  the  fly,  not  such 
as  require  to  be  pursued  and  pondered;  amusement  is 
preferred  to  instruction;  it  is  easier  to  read  newspapers 
or  novels  or  history  of  the  romantic  type  or  even  phi- 
losophy of  the  verbalistic  variety  than  to  acquire  solid 
knowledge;  it  is  easier  to  feel  the  galvanic  effect  of  a 
poem  than  to  discern  the  beauty  and  feel  the  inspiration 
of  a  scientific  work;  and  far  easier  to  acquire  the  lighter 
lingo  of  knowledge  sufficient  for  the  dabbling  conversa- 
tion of  a  "smoker"  or  an  afternoon  tea  than  it  is  to  think 
and  to  know.  What  I  have  just  now  said  requires  an 
important  qualification, — the  public's  interest  in  science 
can  be  greatly  improved  if  those  who  are  expert  in  a 
branch  of  science  will  teach  those  who  are  not, — but  such 
teaching  has  been  very  slight. 

I  have  just  now  alluded  to  "precision  of  thought" 
and  "the  exercise  of  wit" — "genuine"  wit.  Perhaps  you 
will  allow  me  to  digress  a  little  in  this  connection.  A 
short  while  ago  I  read  a  review,  by  a  distinguished  man 
of  letters,  of  Professor  George  Santayana's  Character 
and  Opinion  in  the  United  States.  The  reviewer  tells  us 
that  the  work  has,  besides  other  excellences,  the  qualities 
of  precision,  wit,  and  beauty.  I  have  read  much  of  San- 
tayana's writing,  including  the  poems  and  the  five  volumes 
of  The  Life  of  Reason.  Undoubtedly,  his  writing  is 
beautiful — that  is  why  I  have  read  it — and  it  is  bright, 
too,  sparkling,  and  full  of  surprises;  perhaps  we  may  say 
that  it  has,  in  one  sense  of  the  term,  wit  also;  to  me  its 
wit  appears  to  be  scintillation  rather  than  genuine  wit  for 
in  this  latter  there  is  an  element  of  gravity  which  Santa- 


HYPERSPACES  321 

yana  lacks;  at  all  events  if  he  have  wit,  it  is  not  wit  in 
the  sense  in  which  that  quality  is  found  in  Kant,  for  ex- 
ample, or  Hume  or  Spinoza  or  Descartes  or  Pascal  or 
Aristotle.  As  for  the  quality  of  precision,  it  is  certain 
that  neither  Santayana  nor  his  reviewer  has  it  or  indeed 
knows  the  meaning  of  it  as  it  is  rightly  understood  by 
logicians  or  mathematicians,  for  the  writers  in  question 
are  not  logicians.  They  do  indeed  produce  literature — 
beautiful  literature — but  it  does  not  belong  to  the  litera- 
ture of  knowledge,  which  is  also  beautiful;  the  literature 
it  belongs  to  is  the  literature  of  opinion,  some  of  which 
is  not  even  beautiful,  for  though  it  includes  such  beautiful 
writing  as  that  of  Benedetto  Croce,  for  example,  yet  it 
embraces  work  like  Professor  Bliss  Perry's  The  Present 
Conflict  of  Ideals,  which  we  must  allow  is  a  kind  of  lit- 
erature even  though  it  remind  one  of  a  traveling  sales- 
man displaying  his  wares  or,  less  dimly,  of  an  indiscrimi- 
nate "feeder"  who  loves  to  talk  of  the  things  he  has 
tasted  and  who  sometimes  ascribes  to  bad  food  or  bad 
cuisine  distress  that  is  due  to  enfeebled  or  feeble  digestion. 
Indeed  Professor  Santayana,  like  his  reviewer,  is, 
primarily  and  essentially,  a  poet;  but  there  are  three  kinds 
of  poetry:  there  is  the  poetry  of  pure  thought, — the 
poetry  of  Logic, — and  there  are  two  other  kinds — the 
hypological  and  the  hyperlogical.  Each  of  the  kinds  has 
a  muse  of  its  own;  that  of  the  first  kind,  as  I  said  in  a 
previous  lecture,  is  called  Logical  Rigor,  an  austere  god- 
dess, guardian  of  precision,  mistress  of  the  silent  har- 
monies of  perfect  thought.  By  that  muse  poets  like 
Santayana  have  not  been  inspired. 

Returning  from  the  digression,  let  us  now  endeavor 
to  answer  the  main  question  of  this  lecture:  what  is  the 
meaning   of   the   term    "hyperspace"    or   "//-dimensional 


322  MATHEMATICAL   PHILOSOPHY 

space,"  where  n  is  greater  than  three?  I  have  said  "the 
meaning"  as  if  there  were  only  one.  As  a  matter  of  fact, 
the  term  has  three  meanings,  which,  though  they  are 
closely  related,  it  is  essential  to  distinguish  if  we  are  to 
avoid  confusion.  I  shall  try  to  explain  them  clearly,  re- 
serving the  most  interesting  one  for  the  last.  Certain 
refinements  of  refinements  I  shall  avoid  as  likely  to  ob- 
scure and  hinder  rather  than  to  clarify  and  help  in  a  first 
presentation.  It  will  be  very  advantageous  as  a  pre- 
liminary to  speak  of  some  simple  matters  connected  with 
the  system  of  real  numbers. 

There  is,  as  you  doubtless  know,  an  extensive  and  very 
refined  theory  of  the  logical  genesis  and  properties  of  these 
numbers.  The  theory  is  to  be  found  in  the  numerous 
works  dealing  with  functions  of  a  real  variable,  the 
profoundest  treatment  of  the  subject  being  that  in  the 
Principia  Mathematica.  I  am  not  going  to  assume  that 
you  are  familiar  with  that  theory.  I  take  it  for  granted, 
however,  that  you  are  sufficiently  acquainted  with  the 
system  of  real  numbers  to  understand  fairly  well  what  I 
purpose  to  say  about  it  here.  You  are  aware  that  the 
system  is  composed  of  the  infinitude  of  positive  and 
negative  integers,  the  infinitude  of  rational  fractions,  and 
the  infinitude  of  irrational  numbers  like  V2,  for  example, 
and  including  the  so-called  transcendental  numbers  such, 
for  example,  as  the  familiar  specimens,  -w  and  e.  His- 
torically these  numbers  were  called  "  real  "  to  distinguish 
them  from  the  so-called  "  imaginary "  numbers,  like 
\/  —  2,  for  example,  which  latter  were  long  regarded,  quite 
unjustly,  as  an  ungenuine  kind  of  number.  Today  the 
old  adjectives,  real  and  imaginary,  are  still  regularly 
employed,  but  they  no  longer  signify  that  the  numbers 


HYPERSPACES  323 

thus  designated  are  either  more  or  less  genuine  than  other 
numbers. 

The  system  of  real  numbers  is  a  vast  system  or  class — 
the  multitude  of  the  numbers  in  it  is  indeed  very  great. 
The  extent  of  the  multiplicity, — the  number,  if  you  please 
of  all  the  real  numbers, — is  often  conveniently  denoted  by 
the  familiar  symbol  for  infinity,  oo .  The  same  symbol 
is  used  to  indicate  how  many  things,  elements  or  members 
there  are  in  any  other  equally  numerous  system  or  class — 
say,  that  of  the  points  in  a  straight  line.  If  a;  be  a  vari- 
able representing  "  any  one  "  of  the  real  numbers,  we  say 
that  x  has  one  degree  of  freedom;  we  say  also  that  the 
system  of  real  numbers  is  a  o;z<"-dimensional  system  or 
class;  in  like  manner  we  say  that  a  point  P,  if  free  to 
move  along  a  straight  line,  has,  in  virtue  of  that  fact,  one 
degree  of  freedom,  and  that  a  line,  regarded  as  a  system 
of  points,  has  one  dimension  or  is  ow^-dimensional. 

Now,  a  real  number  is  one  thing  and  a  pair  of  them  is 
another.  Such  pairs  constitute  a  system  (of  pairs).  How 
many  pairs  are  in  the  system?  It  is  easy  to  tell.  Let 
the  symbol  (x,  y)  be  a  variable  representing  any  one  of 
the  pairs;  give  y  some  definite  value,  say,  yi,  and  let  x 
vary — x  can  take  oo  values;  each  of  these  taken  with  y\ 
gives  a  pair,  and  so  we  get  oo  pairs;  in  each  of  these  we 
may  replace  y  1  by  any  of  the  oo  of  values  that  y  may  take; 
we  so  obtain  in  all,  as  you  see,  oo  times  oo ,  or  oo  2,  pairs. 
It  is  plain  that  the  variable  (x,  y),  an  arbitrary  or  unde- 
termined pair  of  the  system,  has  two  degrees  of  freedom, 
owing  to  the  variability  of  the  two  parameters,  or  coor- 
dinates, x  and  y.  Thus  the  system  of  all  the  pairs  of  real 
numbers  is  a  /wo-dimensional  system  (of  pairs).  You  see 
at  once  that  the  system  of  triads  or  triplets  of  real  numbers 
has  three  dimensions,  or  is  a  tri-dimensional  system  (of 


324  MATHEMATICAL    PHILOSOPHY 

triads);  that  the  system  contains  oo3  triads;  and  that  an 
undetermined  triad  (x,  y,  z)  enjoys  three  degrees  of  free- 
dom, one  for  each  of  the  mutually  independent  coordinates, 
or  parameters,  x,  y  and  z.  The  generalization  is  obvious 
and  easy;  if  we  think  of  the  system  of  all  the  sets 
(#1,  #2,  •  •  •  >  *n)  of  real  numbers,  each  set  composed  of  n 
numbers,  it  is  readily  evident  that  the  system  is  an 
^-dimensional  system  (of  sets);  that  it  contains  oon  sets; 
and  that  a  free,  or  undetermined,  set  of  the  system 
possesses  n  degrees  of  freedom. 

I  have  been  speaking  mainly  of  numerical  things.  I 
am  now  going  to  speak  of  things  geometrical  or  spatial, 
and  will  first  make  a  little  more  precise  what  I  said  a 
moment  ago  regarding  a  line.  Let  I  be  a  straight  line; 
choose  a  point  of  L  for  origin  of  distances  and  mark  it  0; 
let  P  denote  any  point  of  L;  denote  by  x  the  distance  (in 
terms  of  some  chosen  unit)  from  0  to  P,  it  being  under- 
stood that  x  is  positive  or  negative  according  as  P  is  on  the 
one  side  of  0  or  on  the  other— of  course  x  will  be  zero  if 
P  coincide  with  0.  Thus  a  reciprocal  one-to-one  corre- 
spondence is  set  up  between  the  real  numbers  of  the 
system  thereof  and  the  points  of  L;  x  represents  P  numer- 
ically, P  represents  x  geometrically;  and  x  is  the  coor- 
dinate or  parameter  of  P.  Notice  that  L  is  here  the  field 
or  the  space  of  operation  and  that  we  are  regarding  it  as  a 
field  or  a  space  of  points.  In  the  light  of  the  preliminary 
discussion  respecting  numbers,  you  see  immediately  that 
a  straight  line,  regarded  as  a  space  of  points,  is  one- 
dimensional,  since  it  perfectly  matches,  as  indicated,  the 
one-dimensional  system  of  the  real  numbers.  In  other 
words  a  line  is  a  point-space  of  one  dimension.  You 
catch  the  idea:  if  a  point  of  a  line  depended  upon  two  or 
more  coordinates  instead  of  only  one,  we  should  say  that  a 


HYrERSPACES  32.5 

line  is  a  point-space  of  two  or  more  dimensions.  All  this 
is  obvious.  I  have  stressed  it  because  it  is  essential  to  a 
good  understanding  of  any  one  of  the  meanings  of  hyper- 
space.  Let  us  turn  to  analogous  considerations  in  the 
case  of  a  plane. 

In  Lecture  V  we  saw  that,  by  means  of  a  pair  of 
rectangular  axes  and  a  unit  of  length,  a  one-to-one  corre- 
spondence can  be  established  between  the  points  P  of  a 
plane  and  the  pairs  (x,  y)  of  real  numbers.  The  pair 
represents  the  point,  and  the  point  the  pair;  the  x  and  y 
of  a  pair  are  at  once  the  coordinates  of  the  pair  and  of  the 
corresponding  point.  There  are  as  many  points  in  the 
plane  as  there  are  pairs  of  real  numbers  in  the  system  of 
such  pairs.  Hence  a  plane  contains  oo2  points;  a  point 
that  is  free  to  move  in  a  plane  and  is  confined  thereto 
has  two  and  but  two  degrees  of  freedom;  a  plane,  regarded 
as  a  space  of  points,  is  a  /z#o-dimensional  space;  and  you 
see  why  it  is  so  called,— it  is  because  the  points  of  a  plane 
match,  in  one-to-one  fashion  as  we  have  seen,  the  pairs  of 
real  numbers  in  the  two-dimensional  system  of  such  pairs. 

And  now  what  shall  we  say  of  ordinary  space?  What, 
I  mean,  shall  we  say  of  that  immense  region  or  room  in 
which  we  are  immersed  and,  with  us,  our  floating  world 
and  the  stars?  Let  us  think  of  it  as  a  field,  or  a  plenum, 
of  points.  What,  then,  is  its  dimensionality?  It  is  easy 
to  ascertain. 

Choose  three  mutually  perpendicular  planes;  they 
have  a  common  point  0,  called  the  origin;  they  determine 
three  lines,  OX,  OY,  OZ,  called  axes;  agree  that  a  distance 
measured  parallel  to  an  axis  shall  be  positive  or  negative 
according  as  it  is  reckoned  in  the  sense  of  the  arrow  or  in 
the  opposite  sense;  note  that  the  three  planes  divide  the 
whole  of  space  into  eight  compartments;    choose  a  unit 


326  MATHEMATICAL   PHILOSOPHY 

of  length;  let  P  be  any  point  and,  as  in  the  figure,  denote 
its  distances  from  the  coordinate  planes  by  x,  y  and  z, 
called  the  coordinates  of  P;  they  are  also  the  coordinates 
of  the  triad  (x,  y,  z).     It  is  plain  that  a  P  determines  a 


Qcyz) 


->X 


Fig.  29. 

triad,  and  a  triad  a  P.  You  see  at  once  that  ordinary 
space,  if  regarded  as  a  plenum  of  points,  has  three  dimen- 
sions, since  the  points  match  the  triads  in  the  three- 
dimensional  system  of  triads  of  real  numbers.     In  such 


HYPERSPACES  887 

space  there  are,  you  see,  oo3  points,  and  a  point  has  therein 
three  degrees  of  freedom  and  only  three. 

We  are  going  very  soon  to  see  very  clearly  one  of  the 
three  meanings  of  the  term  n-dimensional  space,  n  greater 
than  three.  Do  not  fail  to  note  that  thus  far  we  have 
regarded  the  line,  the  plane,  and  ordinary  space  as  fields, 
or  plena,  or  spaces,  of  points;  the  point,  that  is,  has  been 
taken  for  element;  but  nothing  constrains  us  to  elect  the 
point  to  that  position;  we  can  geometrize  just  as  well, 
sometimes  better,  with  some  other  entity  taken  as  element; 
we  may  choose  for  element  the  point  -pair  or  point  triad, 
and  so  on;  in  the  case  of  the  plane,  we  may  take  the 
line  or  the  circle  or  something  else  for  element;  in  the 
case  of  ordinary  space  we  may  take  for  element  any  of 
the  foregoing  entities  or  a  plane,  for  example,  or  a  sphere, 
and  so  on.  It  is  true  that,  from  time  immemorial  until 
a  little  less  than  a  century  ago,  the  point  was  exclusively 
employed  as  geometric  or  spatial  element,  but  there  is 
nothing  in  the  ten  commandments  nor  even  in  the  Vol- 
stead Act  to  prevent  the  use  of  something  else.  The 
point's  ages-old  monopoly  was  broken  up  mainly  by 
Julius  Pliicker  (1801-1868) — one  of  the  greatest  of 
geometricians  and  a  distinguished  physicist  besides, — 
who  geometrized  the  plane  in  terms  of  its  lines  and 
geometrized  ordinary  space  in  terms  of  its  planes  and  its 
lines — thus  emancipating  geometry  forever  from  its  old 
bondage  to  points.  A  geometry  in  which  the  point-^xztr 
is  taken  for  element  will  deal  with  the  properties  of 
configurations  composed,  not  of  points,  but  of  point-pairj-. 
Now,  in  a  line  a  po\nt-pair  has  two  coordinates,  two 
degrees  of  freedom;  in  a  plane  it  has  four;  in  ordinary 
space  it  has  six;  a  line  has  00 2  point-pairs;  a  plane,  00 4 
of  them;    and  ordinary  space,  00  6;    and  so  you  see  that, 


328  MATHEMATICAL   PHILOSOPHY 

regarded  as  a  plenum  or  space  of  point-pairs,  a  line  has 
two,  a  plane  jour,  and  ordinary  space  six,  dimensions. 
You  see  that  the  dimensionality  of  a  space  depends,  not 
only  upon  the  space  itself,  but  also  upon  the  entity 
employed  as  element.  You  see  easily  that,  in  respect  to 
point-triads,  the  dimensionality  of  a  line  is  three,  that  of 
a  plane  is  six,  and  that  of  ordinary  space  is  nine;  and 
you  see  that,  if  we  take  for  element  the  point-j^  contain- 
ing n  points,  then  a  line  is  an  w-dimensional  space,  a  plane 
has  2ft  dimensions,  and  the  dimensionality  of  ordinary 
space  is  3«.  There  is  no  sense  in  simply  saying  that  a 
plane,  for  example,  or  that  ordinary  space  has  such-and- 
such  a  dimensionality  (or  number  of  dimensions);  what 
we  have  to  say  is  that  it  has  such-and-such  a  dimen- 
sionality when  it  is  conceived  as  a  space  or  plenum  of 
elements  of  such-and-such  a  kind.  When  people  say 
simply,  as  they  often  do,  that  space  (meaning  ordinary 
space)  has  three  dimensions,  they  mean — though  they 
do  not  know  well  what  they  mean — that  it  has  three 
dimensions  as  a  space  of  points.  If  you  think  they  know 
what  they  mean,  ask  them  what  they  mean.  For  addi- 
tional examples  showing  that  space  dimensionality  de- 
pends upon  space  element,  consider  the  following.  In 
a  previous  lecture  we  saw  that  in  a  plane  a  line  has  two 
coordinates,  two  degrees  of  freedom — a  plane  being  pre- 
cisely as  rich  in  lines  as  in  points;  and  so  a  plane  is  a  two- 
dimensional  space  of  lines,  as  it  is,  we  have  seen,  of  points. 
What  is  the  line  dimensionality  of  ordinary  space?  It  is 
easily  seen  to  be  four.  To  see  it,  reflect  that  a  line  is 
determined  by  two  points,  say  a  point  in  the  plane  of  the 
floor  of  this  room  and  a  point  in  the  plane  of  the  ceiling; 
each  of  the  points  (kept  in  its  plane)  has  two  coordinates, 
two  degrees  of  freedom,  and  so,  you  see,  the  line  has  four. 


HYPERSPACES  329 

To  distinguish  a  line  of  ordinary  space  from  all  its  other 
lines,  it  is  necessary  and  sufficient  to  tell  four  independent 
facts  about  it;  ordinary  space  contains  oo4  lines,  and  you 
see  that  Pliicker's  famous  line  geometry  (of  ordinary 
space),  which  studies  configurations  composed  of  lines 
(and  not  of  points),  is  a  four-dimensional  geometry.  Let 
us  return  for  a  moment  to  the  plane;  think  of  it  as  a 
plenum  of  circles.  Each  of  its  points  is  the  center  of  an 
oo  of  circles,  and  it  has  oo2  points;  and  so,  you  see,  a  plane 
has  oo3  circles;  in  a  plane  the  circle  has  three  degrees  of 
freedom, — three  coordinates  or  parameters;  a  plane  of 
circles  is  a  /Am'-dimensional  space — as  rich  in  circles  as  in 
point-triads — as  rich  in  circles  as  ordinary  space  in  points. 
You  can  readily  show  that  ordinary  space  is  four-dimen- 
sional in  spheres,  as  we  have  seen  it  to  be  in  lines,  five- 
dimensional  in  flat  \ine-pencils  (explained  before),  six- 
dimensional  in  circles,  and  so  on  and  on  to  your  heart's 
content. 

I  venture  to  believe  that  the  foregoing  illustrations 
have  sufficiently  disclosed  one  of  the  meanings  of  the 
term  "  hyperspace  ":  that  meaning,  namely,  according 
to  which  the  term  signifies  an  ensemble  of  geometric,  or 
spatial,  entities,  or  elements,  of  such  a  kind  that  an 
undetermined  (or  arbitrary)  one  of  them  has,  in  the 
ensemble,  four  or  more  degrees  of  freedom.  This  state- 
ment is  not  designed  to  be  a  definition  of  the  meaning, 
but  only  a  good-enough  description  of  it.  In  this  meaning 
of  hyperspace  there  certainly  is  nothing  to  mystify;  for, 
in  order  to  find  examples  of  such  hyperspaces,  one  is  not 
obliged  to  perform  the  familiar  mathematical  feat, — which 
many  good  people  seem  to  find  difficult  or  even  impossible, 
—of  going  beyond  the  great  domain  of  Imagination  into 
the  infinitely  vaster  domain  of  pure  Conception.     I  have 


330  MATHEMATICAL   PHILOSOPHY 

sometimes  felt  that  no  student  is  intellectually  fit  to  be 
graduated  from  college  who  does  not  easily  and  habitually 
recognize  the  immense  and  fundamental  difference  between 
those  domains.  Such  a  student  is  as  meagrely  disciplined 
as  one  who  believes  that,  if  two  things  or  persons  be  each 
of  them  indispensable,  they  are  therefore  of  equal  impor- 
tance— a  rank  fallacy  vitiating  90  per  cent  of  current  social 
philosophy  throughout  the  world.  I  once  heard  a  railway 
section-hand  argue  that,  because  his  work  was  indispen- 
sable, he  was  just  as  important  as  the  railway  president. 

Have  you  observed  that  among  the  hyperspaces  which 
we  have  so  far  taken  occasion  to  notice  there  is  no  hyper- 
space  of  points?  The  examples  have  been  hyperspaces  of 
pomt-pairs  or  of  point-triads,  ...  or  of  n-sets  (of  points), 
...  or  of  lines  or  of  circles  or  of  \me-pencils  or  of  spheres, 
and  you  are  now  doubtless  prepared  to  extend  the  list  of 
such  examples  indefinitely,  for  "  the  clue,  grown  familiar 
to  the  hand,  lengthens  as  we  go  and  never  breaks." 
But  it  is  not  this  kind  of  hyperspace  that  mystifies  the 
layman.  What  he  desires  to  have  you  make  clear  to  him, 
— though  he  may  not  be  able  to  say  so  very  clearly,— is  the 
conception  of  a  hyperspace  of  points.  When  he  asks  you 
to  "  explain  the  fourth  dimension,"  he  is  really  asking 
you  to  explain  the  idea  of  a  space  that  is  4-dimensional 
in  points  in  the  sense  in  which  ordinary  space  is  3-dimen- 
sional  in  points,  a  plane  2-dimensional  and  a  line  i-dimen- 
sional.  And  so  we  see  that  our  further  task  is  thus 
defined.  I  have  said  that  hyperspace  has  three  intimately 
related  meanings.  One  of  them  has  been  explained. 
The  other  two  attach  to  the  term,  hyperspace  of  points, 
or — what  is  tantamount — point-space  of  four  or  more 
dimensions.  And  we  are  now  to  see  what  the  meanings 
are.     They  are  not  hard  to  see  if  we  but  look  attentively. 


HYPERSPACES  331 

Let  us  begin  very  simply  by  recalling  the  fact  that  in  a 
line  a  point  is  represented  by  one  coordinate  x,  in  a  plane 
by  a  pair  (x,  y)  of  them,  and  in  ordinary  space  by  a  triad 
(x,  y,  z)  of  them.  Now,  instead  of  always  saying  that  the 
point  is  thus  "  represented,"  it  is  very  common,  because 
very  convenient,  to  say  that  the  x  or  the  (x,  y)  or  the 
(x,  y,  z)  is  the  point,  and  this  is  done,  explicitly  or  implic- 
itly, in  very  many  ways;  thus  we  say,  for  example, 
"  consider  the  point  (x,  y)  "  or  "  consider  the  line 
ax-\-by-\-c=0  "  instead  of  saying,  "consider  the  system 
of  those  pairs  of  values  of  x  and  y  which  satisfy  the 
equation  ax-\-by+c=0."  This  familiar  way  of  speaking 
as  if  real  numbers,  pairs  thereof  and  triads  thereof  were 
indeed  points  and  as  if  equations  were  indeed  loci,  is  very 
brief,  very  neat  and  very  stimulating,  too,  on  account 
of  its  keeping  the  mind  continually  delighted  with  the 
presence  of  geometric  or  spatial  imagery.  You  see  that, 
in  order  to  be  thoroughly  consistent  in  this  manner  of 
speaking,  we  should  have  to  say  that  the  system  of  real 
numbers  x  is  the  line,  that  the  system  of  pairs  (x,  y)  is 
the  plane,  and  that  the  system  of  triads  (x,  y,  z)  is 
(ordinary)  space. 

And  now  what  I  wish  to  point  out  is  that  just  such  a 
thoroughgoing  geometric  way  of  speaking  is  often  em- 
ployed by  mathematicians  when  dealing  with  four  or 
more  real  variables,  x,  y,  z,  zv,  etc.  That  is  to  say,  if  they 
be  handling  four  variables,  they  call  a  tetrad  (x,  y,  z,  zv) 
of  numbers  a  point  and  the  totality  of  such  points  they 
quite  consistently  call  a  point-space  of  four  dimensions. 
And  in  like  manner  for  any  yet  larger  number  of  variables. 

Query:  when  mathematicians  thus  speak,  do  they  sup- 
pose that  there  exists  a  4-dimensional  space  containing 
points  for  the  number  tetrads  (x,  y,  z,  zv)  to  represent  as 


332  MATHEMATICAL    PHILOSOPHY 

there  exists  a  plane  (say)  containing  points  for  the  dyads 
(x,  y)  to  represent?  The  answer  is  that  some  of  them 
suppose  it  and  some  of  them  do  not;  and  in  this  fact  is 
the  key  to  the  two  meanings  of  the  term,  "  hyperspace 
of  points."  According  to  one  of  the  meanings,  a  point- 
space  of  w-dimensions  is,  strictly  speaking,  not  a  space 
at  all,  but  is  simply  and  purely  an  w-dimensional  system 
of  number  sets  (each  having  n  numbers);  and  the  theory 
or  science  of  such  a  system,  verbally  geometrized  as  I  have 
indicated,  is  not  genuine  geometry,  but  is  simply  a  species 
of  ^-dimensional  Algebra  or  Analysis  conducted  and 
couched  in  geometric  speech.  Such,— to  take  an  example 
as  early  as  1847, — is  Cauchy's  Memoir  sur  les  lieux 
analytiques  where  he  says,  "  We  shall  call  a  set  of  n 
variables  an  analytic  point,  an  equation  or  system  of 
equations  an  analytical  locus,"  and  so  on.  According  to 
the  other  meaning  a  hyperspace  of  points  is  held  to  be  a 
genuine  space;  the  points  constituting  it,  though  repre- 
sentable  by  number  sets  of  n  numbers  each,  are  distinct 
from,  and  independent  of,  such  sets  as  the  points  of 
ordinary  space  are  distinct  from,  and  independent  of,  their 
representative  number  triads  (x,  y>  z) ;  and  the  theory  of 
such  a  space,  whether  the  theory  be  built  up  synthetically 
or  analytically,  is  genuine  w-dimensional  geometry — 
geniune  geometry  of  a  hyperspace  of  points. 

I  am  sure  that  in  this  connection  you  are  impatient  to 
raise  the  question  whether  hyperspaces  of  points  may  be 
said  to  exist,  and,  if  we  allow  that  they  may,  in  what 
sense  of  the  term  "  exist."  The  question  evidently 
involves  nice  matters  both  of  psychology  and  of  meta- 
physics. Many  mathematicians  have  not  carefully  con- 
sidered those  "  nice  matters  "  and  are  quite  content 
(because  of  convenience   as   already   explained)  to  speak 


HYTERSPACES  333 

as  if  the  hyperspaces  in  question  exist,  without  thereby 
intending  either  to  affirm  or  to  deny  that  they  do  exist 
in  fact.  After  much  reflection  I  have  myself  no  longer 
any  doubt  in  the  premises,  and  in  my  Human  Worth  of 
Rigorous  Thinking,  p.  256,  I  have  stated  my  conviction 
in  the  words:  hyperspaces  have  every  kind  of  existence  that 
may  be  warrantably  attributed  to  the  space  of  ordinary 
geometry.  The  considerations  that  have  led  me  to  that 
conclusion  are  set  forth  at  sufficient  length  in  the  work 
cited  and  need  not  be  restated  here. 

In  relation  to  the  matter  in  hand,  note  carefully  the 
sharp  difference  of  temper,  attitude  and  interest  between 
the  following  two  classes  of  mathematicians:  those  of  the 
one  class,  primarily  interested  in  geometry,  affirm  the 
existence  of  a  point-space  of  n  dimensions  and  then  inves- 
tigate its  properties — build  up  its  geometry — by  the 
algebraic  or  analytic  method — by  applying,  that  is,  the 
theory  of  n  independent  numerical  variables  (say,  x\,  X2, 
X3,  .  .  .  ,  xn)  to  the  postulated  space;  those  of  the  other 
class,  primarily  interested  in  algebra  or  analysis,  employ, 
in  their  discourse  about  the  system  of  n  variables,  the 
nomenclature  of  the  geometry  of  a  point-space  of  n  dimen- 
sions as  if  there  were  such  a  space,  but  do  not  affirm  its 
existence.  (Of  course  the  former  class  are  not  obliged 
to  employ  the  analytic  method  nor  are  the  latter  class 
obliged  to  employ  geometric  speech.)  Note  the  "  as  if  " 
in  the  following  extract  from  J.  J.  Sylvester's  "  A  Plea 
for  the  Mathematician  "  (Mathematical  Papers,  Vol.  II): 

Dr.  Salmon  in  his  extension  of  Chasles'  theory  of 
characteristics  to  surfaces,  Mr.  Clifford  in  a  question 
of  probability,  and  myself  in  my  theory  of  partitions, 
and  also  in  my  paper  on  barycentric  projection,  have  all 
felt    and    given    evidence    of  the    practical    utility    of 


334  MATHEMATICAL    PHILOSOPHY 

handling  space  of  four  dimensions  as  if  it  were  con- 
ceivable space. 

By  "  conceivable  "  he  here  means  actual. 

It  is  noteworthy  that  in  the  difference  between  affirm- 
ing the  existence  of  hyperspaces  of  points  and  merely 
speaking  as  if  such  spaces  existed  we  have  a  striking 
illustration  of  the  Kantian  distinction  between  postulating 
and  feigning—between  hypothesis  and  fiction — a  much- 
neglected  distinction  justly  and  stoutly  insisted  upon  by 
Vaihinger  in  his  great  Philosophie  des  Als  Ob  as  being  of 
fundamental  importance  in  the  philosophy  of  science  and 
the  philosophical  history  of  thought  in  general.  The 
distinction  is  indeed  very  important  and  very  wide  in  its 
application.  One  must  be  pretty  dull  not  to  perceive 
that  the  difference  is  radical  between  saying,  for  example, 
"  there  is  an  infinite  and  all-wise  God  and  hence  we 
ought  to  live  so-and-so  "  and  saying  "  we  ought  to  live 
as  if  there  were  such  a  God  ";  or  between  saying  "  there 
is  a  universal  ether  having  such-and-such  properties  and 
that  is  why  light  behaves  in  such-and-such  a  way  "  and 
saying  "  light  behaves  as  if  there  were  an  ether  having 
such-and-such  properties  ";  or  between  saying  "  the 
human  soul  is  immortal  and  hence  we  ought  to  live  so- 
and  so  "  and  saying  "  we  ought  to  live  as  if  the  human  soul 
were  immortal  ";  and  so  on  throughout  the  whole  range  of 
thought.  A  postulate  or  hypothesis,  as  here  understood, 
is  a  proposition  and  is  true  or  false;  but  a  fiction  is  not  a 
proposition  and  is  neither  true  nor  false.  It  would  be 
very  enlightening  to  make  a  survey  of  scientific  "  hy- 
potheses "  with  a  view  to  ascertaining  which  of  them  are 
genuine  hypotheses  and  which  ones  are  only  fictions — 
only  as  ifs.  There  can  be  no  doubt  that  many  a  scientific 
worker  would    be    astonished    at   the   results   of  such    a 


HYPERSPACES  335 

critical  survey.  An  excellent  clue  to  Vaihinger's  work  is 
found  in  Dr.  W.  B.  Smith's  penetrating  review  of  it  in 
The  Literary  Review  (N.  Y.  Evening  Post),  July  9,  1921. 

The  hyperspaces  of  points  are  unimaginable  worlds — 
unimaginable  for  us  humans,  I  mean,  in  our  present  stage 
of  development — but  they  are  thoroughly  conceivable 
worlds;  and  for  mathematical  purposes  nothing  is 
demanded  but  thorough  conceivability.  The  importance 
of  that  fact  is  fundamental.  Experience  has  taught  me 
that  it  is  hard  to  drive  the  fact  home  to  the  average 
understanding.  Wherever  the  distinction  involved  in  the 
fact  is  not  understood  by  "  critics,"  whether  scientific 
or  literary  or  philosophic,  criticism  is  blind  and  worse 
than  futile,  being  at  once  misled  and  misleading,  confused 
and  confusing.  It  is  important  to  observe  and  to  bear 
in  mind  that,  with  respect  to  the  great  powers,  or  types, 
of  mental  activity, — Sensibility  (or  Sense-perception), 
Imagination,  Conception, — we  humans  fall  into  three 
classes:  there  are  those  who  have  the  first  power  but 
little  of  the  second;  there  are  those  who  have  the  first  and 
the  second  powers  but  little  of  the  third;  and  there  are 
those  who  have  in  good  measure  the  three  powers.  The 
second  class  is  related  to  the  third  very  much  as  the  first 
is  related  to  the  second.  Beware  of  the  first  two  classes, 
— they  can  give  you  neither  science  nor  genuine  philosophy 
nor, — properly  speaking, — criticism.  Are  they  aware  of 
their  limitations?  No;  at  all  events  not  keenly.  How 
could  they  be? 

There  are  various  avenues  by  which  beginners  may 
approach  those  unimaginable  worlds  and  enter  them;  and 
that  is  a  blessing,  for  the  worlds  are  replete  with  wonders. 
One  of  the  ways  is  that  followed  by  Professor  H.  P.  Man- 
ning, for  example,  in  his  Geometry  of  Four  Dimensions, 


336  MATHEMATICAL   PHILOSOPHY 

the  reading  of  which  requires  no  more  preparatory  knowl- 
edge of  geometry  and  geometric  method  than  can  be 
acquired  in  a  good  high  school. 

Another  of  the  ways  is  the  deliberate  and  patient  way 
of  postulate  procedure.  I  am  not  going  to  take  the  time 
that  would  be  necessary  to  spread  before  you  here  a 
system  of  postulates  for  the  geometry  of  a  point-space  of 
n  dimensions.  To  do  so  would  be  wasteful,  for  I  miy 
assume  that  you  are  now  pretty  familiar  with  one  or 
more  postulate  systems  for  a  space  of  three  dimensions, — 
as  that  of  Hilbert  or  that  of  Veblen  or  that  of  Veblen 
and  Young  (for  3-dimensional  projective  geometry), — 
and  only  slight  alteration  of  any  such  system  is  needed  to 
convert  it  into  a  system  available  for  4,  5  or  n  dimensions. 
I  will  content  myself  with  referring  you  to  page  24  of 
Veblen  and  Young's  Projective  Geometry,  for  example,  for 
a  clear  indication,  if  you  require  to  be  shown,  of  the 
simple  sort  of  alteration  that  will  suffice. 

The  two  foregoing  ways  of  working  into  and  working  in 
the  worlds  of  hyperspace  are,  as  you  observe,  the  ways  of 
pure,  or  synthetic,  geometry  as  distinguished  from  ana- 
lytic, or  algebraic,  geometry  (which  latter,  let  me  remind 
you,  is  only  a  geometric  method).  The  "  pure  "  ways  are 
followed  with  especial  frequency  by  the  Italian  geometri- 
cians, though,  of  course,  the  latter  often  employ  the 
analytic  method  also. 

Of  the  latter  method,  I  have  already  said  enough  for 
the  stated  purpose  of  this  lecture.  An  excellent  way  for  a 
beginner  is  found  in  a  somewhat  rough  mixture  of  the 
"  pure  "  and  the  algebraic  ways,  guided  by  the  chief  of  all 
intellectual  guides —  analogy — as  follows : 

Let  I  be  a  line  (say  a  projective  line)  of  points;  it  is 
a  space  of  one  dimension,  Si;    if  you  like,  you  may  say 


HYPERSPACES  337 

that  a  point  is  itself  a  space  of  zero  dimensions,  and  denote 
it  by  So;  conceive  a  point  P  not  in  L  and  consider  the  set 
of  lines  joining  P  to  the  points  of  L;   in  this  set  there  are 

00  lines;  think  of  the  ensemble  of  all  the  points  of  these 
lines;  there  are  oo2  of  them;  they  evidently  constitute  a 
plane — a  2-dimensional  point-space,  S2.  Now  conceive  a 
point  P  not  in  S2  and  think  of  the  set  of  all  lines  joining 
P  to  the  points  of  S2',  of  such  lines  there  are  00 2;  plainly 
the  points  of  these  lines  together  constitute  a  3-dimensional 
space  (like  the  familiar  space  of  ordinary  solid  geometry); 
it  has  00 3  points;  denote  it  by  S3.  In  the  next  step  imag- 
ination ceases  to  accompany  our  thought;  so  much  the 
worse  for  imagination,  for  conception  goes  on  rejoicing, 
quite  as  before;  if  it  could  not  go  on  endlessly,  there  could 
be,  strictly  speaking,  no  science.  You  see,  of  course, 
what  the  next  step  is;  take  it  boldly:  conceive  a  point  P 
not  in  S3;  conceive  P  joined  by  lines  to  all  the  points  of 
S3;  of  such  lines  there  are,  as  you  see,  oo3  in  all;  each  of 
them  has  an  00  of  points  and  all  of  them  together  give 
you  00 4  points  constituting  a  4-dimensional  space,  S4. 
Repetition  of  the  process  yields  S3,  then  Sc,  and  so  on  till 
you  have  the  conception  of  a  point-space  of  any  required 
dimensionality,  however  high. 

Having  once  formed  the  concept  of  hyperspace,  what 
then?  What  is  to  be  done  with  it?  The  answer  depends 
upon  you — upon  your  interest  and  your  ability.  Those 
higher  worlds,  I  have  said,  are  replete  with  wonders. 
These  are  not  (yet)  shown  in  the  "  movies."     Neither  can 

1  exhibit  them  here.  If  you  wish  to  see  them  you  must 
pay  a  certain  price — that  of  seriously  studying  hyper- 
space geometry.  Of  this  geometry  the  literature  is  large, 
is  growing,  and  will  continue  to  grow.  An  excellent  intro- 
duction   to   it    is    Schoute's    Mehr dimension  ale   Geometric 


338  MATHEMATICAL    PHILOSOPHY 

where  the  matter  is  handled  systematically,  elementally 
and  many-sidedly.  Among  the  aims  of  the  course  in 
Modern  Theories  of  Geometry,  which  I  have  given  at 
Columbia  University  for  many  years,  is  that  of  helping 
students  to  acquire  a  working  knowledge  of  n-dimensional 
geometry.  The  importance  of  such  a  knowledge  is  by  no 
means  restricted  to  students  of  so-called  pure  mathematics. 
Indeed,  a  few  years  ago,  I  had  the  honor  to  give  a  series 
of  lectures  on  w-dimensional  geometry  to  a  group  of 
physicists  who  had  found  that  without  some  knowledge 
of  hyperspace  methods,  they  could  not  read  the  literature 
of  their  own  subject,  especially  that  of  the  kinetic  theory 
of  gases.  That  was  before  the  present  relativity  rage, 
which,  as  we  saw  in  a  previous  lecture,  avails  itself  of  the 
idea  of  four-dimensional  space.  It  will  take  but  a  minute 
and  it  will  be  instructive  to  show  why  students  of  gas 
theory  are  now  obliged  to  acquire  some  knowledge  of 
^-dimensional  geometry.  It  is  because  some  of  the  fore- 
most writers  on  the  theory, — J.  H.  Jeans,  for  example, — 
have  adopted  and  elaborated  the  following  considerations. 
Suppose  we  have  a  closed  vessel,  say  a  sphere,  filled  with 
gas.  Let  us  suppose  the  gas  is  composed  of  N  molecules. 
These  are  flying  about  hither  and  thither,  all  of  them  in 
motion.  Think  of  one  of  them;  at  a  given  instant  it  is  at 
a  point  (xy  y,  z) ;  at  the  same  time  it  is  moving  so  that  the 
components  of  its  velocity  along  the  axes  of  reference 
are  (say)  u,  v  and  w\  if  and  only  if  we  know  the  six 
coordinates  of  the  molecule  at  an  instant,  we  know  where 
it  is  and  the  direction  and  rate  of  its  going.  The  N 
molecules  constituting  the  gas  thus  depend,  you  see,  upon 
6N  coordinates.  At  any  instant  these  have  definite  values. 
Together  these  values  define  the  "  state  "  of  the  gas  at 
that    instant.     Now,    say   the   writers    in    question,   the 


HYPERSPACES  339 

6N  values  determine  a  point  in  a  space  of  6N  dimensions. 
Thus  there  subsists  a  correspondence  between  such  points 
and  the  varying  gas  states.  As  the  state  of  the  gas 
changes  (owing  to  the  motions  of  the  molecules)  the 
corresponding  point  generates  a  path,  or  locus,  in  the 
space  of  6N  dimensions;  and  so  the  behavior  or  history  of 
the  gas  (as  a  whole)  gets  geometrically  represented  by 
loci  in  the  mentioned  hyperspace.  That  will  suffice  as  a 
hint  at  what  has  become  a  recondite  mathematical  theory 
— the  kinetic  theory  of  gases. 

I  have  said  that  I  cannot  here  exhibit  the  wonders  to 
be  found  in  the  worlds  of  hyperspace.  To  do  so  in  any 
fair  measure  would  require  many  lectures  as  long  as  this 
one.  I  can  not  refrain,  however,  from  leading  you,  if 
you  be  willing,  to  see  one  of  the  minor  wonders  met  with 
on  the  very  threshold  of  4-dimensional  space.  We  can 
find  it  in  the  "  mixed  way  "  we  were  following  a  little 
while  ago,  guiding  ourselves  by  analogy,  and  at  the  same 
time  you  will  see  how  you  can  yourselves  discover  further 
wonders.  Note  the  facts  carefully  and  note  their  analo- 
gies as  we  start  at  the  bottom  and  ascend  the  scale. 
Observe,  to  begin,  that  in  a  line  (Si)  an  equation  ax+b  =0 
of  first  degree  in  one  variable  (x)  represents  a  point  (So); 
in  a  plane  (S-z)  an  equation  ax-\-by-\-c  =  0  of  first  degree  in 
two  variables  (x,  y)  represents  a  line  (Si),  and  that  two 
such  equations  taken  as  simultaneous  represent  a  point 
(So) — the  common  point  of  the  two  lines;  in  an  ordinary 
space  (S3)  an  equation  ax-\-by-\-cz-\-d  =  0  of  first  degree 
in  three  variables  (xy  y,  z)  represents  a  plane  (SL>),  two 
such  equations  taken  as  simultaneous  represent  a  line 
(Si), — the  line  common  to  the  two  planes, — and  that 
three  such  equations  (if  independent)  together  represent  a 
point  (So) — the  common  point  of  the  three  planes.     You 


340  MATHEMATICAL   PHILOSOPHY 

now  have  the  analogical  clue.  Following  it  you  see  imme- 
diately that  in  a  4-dimensional  space  (S4)  an  equation 
ax+by-\-cz-\-dw+e  =  Q  of  first  degree  in  four  variables 
(x,  y,  z,  w)  represents  an  ordinary  space  (S3) — named  a 
lineoid  by  my  colleague,  Professor  F.  N.  Cole;  that  two 
such  equations  together  represent  a  plane  (S2) — the  plane 
common  to  the  two  lineoids;  that  three  such  equations 
(if  independent)  represent  a  line  (Si) — the  line  common  to 
the  three  lineoids;  and,  finally,  that  four  such  equations 
(if  independent)  represent  a  point  (So)— the  common 
point  of  the  four  lineoids.  You  are  already,  you  see,  in 
the  midst  of  astonishing  things:  you  see  that  an  S4 — a 
hyperspace  of  the  lowest  dimensionality — contains  a  four- 
fold infinity  (00 4)  of  lineoids  (spaces  like  our  own);  you 
see  that  any  two  of  these  have  a  plane  for  their  inter- 
section, that  any  three  independent  lineoids  (in  S4)  have 
a  line  in  common,  and  that  four  of  them  have  one  point 
in  common  and  only  one.  I  spoke  of  showing  you  a 
"  minor  wonder."  It  is  that  in  S4  two  planes  (unless  they 
happen  to  be  in  a  same  lineoid)  have  one  and  only  one 
point  in  common.  To  see  that  this  statement  is  true, 
consider  four  independent  equations  like  the  last  of  the 
foregoing;  two  of  them,  as  we  have  seen,  represent  a 
plane;  the  other  two  represent  another  plane.  What 
points  have  the  planes  in  common?  The  answer  is: 
those  points  whose  coordinates  (x,  y,  z,  w)  satisfy  the 
four  equations.  But,  as  you  know,  such  a  system  of 
equations  is  satisfied  by  only  one  set  of  values.  Hence 
the  proposition.  There  are  many  other  near-lying  mar- 
vels in  S4.  One  of  them  is  that  you  can  pass  from  the 
inside  to  the  outside  of  an  ordinary  sphere  without  going 
through  its  surface.  Another  one  is  this:  if  in  ordinary 
space  you  wish  to  make  a  prison  bounded  by  planes,  you 
have  to  use  at  least  four  planes;  while  in  S4  the  analogous 


HYPERSPACES  341 

prison  is  bounded  by  five  ordinary  SVs.  I  will  mention 
but  one  more.  In  ordinary  space  (S3)  two  planes  have 
but  one  angle;  in  S4  two  planes  make  two  angles  with 
each  other,  so  that,  if  you  would  bring  the  planes  into 
coincidence,  you  must  rotate  one  of  them  about  their 
common  point  in  two  ways.  These  specimens  are  mild 
marvels;  in  S4  their  like  is  inexhaustible;  astonishment 
increases  as  one  ascends  the  summitless  scale  of  dimen- 
sionality, and,  with  astonishment,  also  light  and  edifica- 
tion. Indeed  we  may  say  that  the  science  of  geometry 
is,  properly  speaking,  ^-dimensional  geometry. 

In  closing  this  long  lecture,  I  need  add  but  little  respect- 
ing the  human  significance  of  the  momentous  conception 
with  which  it  has  dealt.  We  have  seen  that,  in  the 
matter  of  scientific  speech,  the  geometry  of  hyperspace 
has  clothed  pure  Analysis  with  the  beauty  and  strength 
of  a  tongue  that  is  at  once  delightful,  stimulating  and 
economical;  we  have  seen  that  the  language,  the  ideas 
and  the  methods  of  w-dimensional  geometry  are  becoming 
rapidly  more  and  more  powerful  agencies  in  the  great 
outlying  domain  of  Physics;  we  have  glimpsed  the  fact 
not  only  that  w-dimensional  geometry  is  in  itself  of  exceed- 
ing great  interest,  but  that  the  geometry  of  the  higher 
worlds  illuminates  that  of  the  lower  as  the  geometry  of 
ordinary  space  illuminates  that  of  the  plane  or  the  line. 
I  have  now,  finally,  to  mention  what  is,  in  my  belief, 
the  chief  consideration.  Human  progress  is  progress  in 
emancipation;  and  in  the  Concept  of  a  summitless 
hierarchy  of  Hyperspaces  is  attained,  I  do  not  say  the 
most  precious,  but  the  amplest,  Freedom  yet  won  by  the 
human  spirit,— room,  I  mean,  for  exterior  representation, 
for  the  architecture,  if  you  please, — of  every  analytic 
doctrine  or  theory,  even  though  there  be  involved  in  its 
structure  an  infinite  number  of  variables. 


LECTURE  XVII 
Non-Euclidean  Geometries 

THEIR   BIRTH   AND   VARIETIES THEIR   LOGICAL    PERFEC- 
TION    THEIR      PSYCHOLOGICAL     DIFFERENCES  

THEIR  SCIENTIFIC  AND   PHILOSOPHIC   SIGNIFICANCE 

ALL    OF    THEM    PRAGMATICALLY    TRUE SCIENCE 

AND    TRAGEDY A    PRELUDE    ON    THE    POPULARIZA- 
TION OF  SCIENCE SCIENCE  AND  DEMOCRACY. 

"In  the  early  part  of  the  last  century  a  philosophic 
French  mathematician,  addressing  himself  to  the  ques- 
tion of  the  perfectibility  of  scientific  doctrines,  ex- 
pressed the  opinion  that  one  may  not  imagine  the  last 
word  has  been  said  of  a  given  theory  so  long  as  it  can 
not  by  a  brief  explanation  be  made  clear  to  the  man 
of  the  street."  x 

The  mathematician  referred  to  is  Gergonne,  one  of 
those  who  assisted  in  the  second  discovery  of  projective 
geometry  (long  after  the  work  of  its  first  discoverer, 
Desargues,  as  I  said  in  a  previous  lecture,  had  been  lost 
and  utterly  forgotten).  It  is  to  Gergonne  that  we  owe 
the  first  enunciation  of  the  great  law  of  Duality — one  of 
the  most  beautiful  and  fertile  principles  of  modern 
geometry.     His  noble  dream  respecting  the  perfectibility 

1  Quoted  from  address  on  Mathematics  written  in  1907  and  published 
as  final  chapter  in  The  Human  Worth  of  Rigorous  Thinking  (Columbia 
University  Press,   1916). 

342 


NON-EUCLIDEAN  GEOMETRIES  343 

of  scientific  theories  ought  to  be  given  in  his  own  words. 
They  are  these: 

On  ne  peut  se  flatter  d'avoir  le  dernier  mot  d'une 
theorie,  tant  qu'on  ne  pent  pas  I'expliquer  en  peu  de 
paroles  a  tin  passant  dans  la  rue. 

Can  the  dream  come  literally  true?  We  are  certain 
that  it  cannot,  for  it  is  an  ideal, — a  genuine  ideal, — and 
genuine  ideals  can  never  be  realized  fully.  Therein  is 
their  precious  value  as  lights  and  lures  of  the  spirit — 
they  are  "ever  flying  perfects,"  not  to  be  overtaken  but 
to  be  pursued  by  us,  as  they  rise  and  soar  and  lead,  for- 
ever. 

The  ideal  of  Gergonne  is  a  democratic  ideal.  To 
pursue  it  is,  therefore,  not  merely  our  privilege;  it  is  a 
great  and  solemn  duty.  Democracy  is  on  trial, — it  is  an 
experiment, — the  greatest  experiment  ever  undertaken  by 
our  humankind.  Unless  the  community  be  pervaded  with 
ever-increasing  scientific  intelligence,  that  supreme  experi- 
ment,— the  sovereign  hope  of  the  world, — is  doomed  to 
failure.  Than  that,  nothing  can  be  more  evident  to  such 
as  reflect.  The  affairs  of  state  must  be  rescued  from  the 
hands  of  ignorant  politicans  and  be  committed  to  scien- 
tific management — to  the  guidance,  that  is,  of  honest 
men  who  know.  That,  too,  is  as  evident  as  anything  can 
become.  How  can  the  destiny  of  the  state  be  committed 
to  the  guidance  of  science  if  the  men  and  women  who 
constitute  the  electorate  know  nothing  of  science,  nothing 
of  its  methods,  nothing  of  its  content,  nothing  of  its 
achievements,  nothing  of  its  spirit,  nothing  of  its  infinite 
potency  for  human  service?  Election  is  selection.  How 
can  the  ignorant  select  the  wise? 

In  view  of  such  considerations,   so  obvious  and  so 


344  MATHEMATICAL    PHILOSOPHY 

important,  it  is  indeed  strange  that  scientific  men  have 
been  so  little   actuated  by  Gcrgonne's  beautiful   dream. 
What  has  been  the  trouble?     What  is  the  secret?     Is  it 
that  scientific  specialists  find  in  the  educated  public  a  lack 
of    scientific    interest?      Tokens    of    scientific    curiosity 
abound  on  every  hand, — witness,  for  example,  the  recent 
world-wide  curiosity  manifested  by  non-specialists  in  the 
theories, — most    recondite   theories, — of   Professor   Ein- 
stein.     No    doubt   such    curiosity   is    often   shallow    and 
transitory,  but  it  can  be  nourished  and  be  thereby  made 
deeper    and    more    enduring.      Do    scientific    specialists 
really  believe   that,    in  general,    educated  non-specialists 
have  not  enough  mind  to  understand  scientific  ideas,  even 
when  these  are  presented  in  non-technical  speech?    If  they 
do,  I  am  convinced  that  they  are  mistaken;  and  if  they 
do,  they  must  be  convinced,  if  they  have  considered  the 
matter,  that  Democracy  is  a  futile  enterprise.    The  same 
conclusion  would  evidently  follow  if  they  held  that,  for 
the  most  part,  scientific  ideas  do  not  admit  of  intelligible 
expression    in    non-technical    terms.       But    they    cannot 
rationally  hold  that  such  expression  is  in  fact  impossible; 
they  may  rightly  regard  it  as  difficult,  as  demanding  the 
patience  and  skill  and  humane  motivity  of  a  special  art, 
but  they  must  know  that  it  is  not  impossible;  for  they 
know  that  scientific  ideas,  however  high  they  be  above 
the  level  of  common  experience  and  common  sense,  yet 
have  their  roots  in  its  homely  soil.     Was  it  Lord  Kelvin 
or  another  sage  who  said  of  mathematics  that  it  is  just 
"common  sense  etherealized"?     The  statement  is  as  true 
of  science  in  general  as  it  is  of  mathematics;  it  is  not  in- 
deed a  complete  characterization  of  either  of  them,  for 
the  process  by  which  they  rise  out  of  common  sense  in- 
volves something  more  than   etherealization,   something 


NON-EUCLIDEAN   GEOMETRIES  345 

more  than  purification,  more  than  elimination  of  dross; 
it  involves,  besides,  a  constructive  process,  a  process  of 
creation.  But,  though  the  statement  is  not  a  complete 
characterization  of  science  in  general  nor  of  any  branch 
thereof,  yet,  regarded  as  a  partial  characterization,  it  is 
fundamentally  true  of  every  branch:  of  all  science  com- 
mon experience, — common  sense, — is  the  basic  soil.  And 
not  only  do  scientific  specialists  know  that  all  scientific 
ideas  have  their  roots  in  the  soil  of  common  sense  but 
they  know,  too,  that  every  single  term  in  the  vast  jargon 
of  science  ultimately  derives  its  meaning,  in  one  way  or 
another,  from  generic  ideas  which,  though  ill  defined  in 
the  common  consciousness,  are  present  there  and  are 
constantly  employed  by  your  man-in-the-street.  The 
process  of  such  derivation  is  a  perfectly  natural  one; 
natural  processes  are,  for  the  most  part,  not  reversible; 
but  this  one  is;  there  is,  I  mean,  no  scientific  idea  what- 
ever, however  complicated  and  refined,  and  there  never 
will  be  one,  that  does  not  admit  of  being  analyzed  and 
ultimately  expressed  in  the  language  appropriate  to  the 
vulgar  elements  whence  the  idea  was  originally  derived. 
In  the  case  of  many  ideas,  such  elemental  analysis  requires 
great  patience  and  skill,  and  their  expression  in  common 
speech  can  not  be  made  perfectly  clear;  it  will  sometimes, 
of  necessity,  be  so  cumbrous  and  prolix  as  to  be  unprofit- 
able except  as  an  exercise;  but  the  thing  can  be  done,  and 
the  point  is  that,  in  an  immense  multitude  of  cases,  it  can 
be  done  in  a  way  to  edify  not  only  the  general  public, 
but  also  the  experts  who  render  the  service. 

The  radical  explanation  of  the  scientific  ignorance  of 
the  educated  public  is  to  be  found  in  the  fact  that,  with 
rare  exceptions,  those  who  understand  do  not  teach — do 
not  teach,  I  mean,  save  in  a  manner  suitable  for  the  train- 


346  MATHEMATICAL   PHILOSOPHY 

ing  of  specialists  like  themselves,  in  terms,  that  is,  that 
are  highly  technical  and  jargonistic.  In  the  course  of  a 
good  many  years  of  university  experience,  I  have  had 
occasion  to  attend  many  public  examinations  of  candidates 
for  the  degree  of  doctor  of  philosophy,  not  only  in  mathe- 
matics but  in  other  branches  of  science.  There  is  one 
question  which  I  have  been  accustomed  to  ask  the  can- 
didates. The  question  is:  Can  you  state  intelligibly,  in 
the  language  current  among  educated  men  and  women 
the  nature  of  your  research, — the  problem  you  have 
solved,  the  methods  you  have  employed,  and  the  results 
you  have  obtained?  And  in  every  instance  the  response 
has  amounted  to  this :  "I  have  never  attempted  to  do 
it;  I  have  not  thought  of  it;  but  I  believe  it  would  be  very 
difficult  or  quite  impossible."  What  is  to  be  said  of  their 
estate  ?  I  think  we  may  say  this :  Their  estate  is  pitiable ; 
they  have  devoted  long  laborious  years  to  qualifying 
themselves  for  a  certain  ordeal, — the  ordeal  of  demon- 
strating that  they  have  acquired  a  certain  competence  of 
highly  technical  scholarship  in  some  field  of  study  and 
that  they  have  the  ability  to  do  independent  research  in 
the  field;  they  have,  let  us  say,  gone  through  the  ordeal 
successfully;  that,  in  itself,  is  well,  but  the  price  they  have 
paid  is  terrible;  they  have  submitted  to  the  painful  process 
by  which  men  are  converted  into  mere  technicians;  the 
education  they  have  acquired  in  the  best  years  of  growth 
is  lacking  in  the  quality  of  amplitude;  they  have  become 
narrowly  technologized;  long  confined  within  the  prison 
walls  of  a  Specialty,  sinking  deeper  and  deeper  in  its 
profound  indeed  but  narrow  shaft,  they  have  become 
more  and  more  detached  from  the  thronging  life  of  the 
world,  and  lost  alike  the  power  of  sympathy  and  the 
power  of  communion  with  their  fellow  men  and  women; 


NON-EUCLIDEAN  GEOMETRIES  347 

they  have  Indeed  qualified  for  membership  in  a  small  and 
insulated  class  of  technicians,  composed,  in  the  main, 
of  spiritually  meagre  men;  and  the  worst  of  it  is  that, 
having  lost  perspective,  they  are  often  vain  of  the  dis- 
tinction; they  are  apt  to  fancy  themselves  investigators, 
and  some  of  them  will  be  but  most  of  them  will  not; 
teaching, — teaching,  I  mean,  in  the  collegiate  sense,  they 
are  prone  to  regard  as  drudgery  with  which  a  cruel  fate 
hampers  their  genius,  while  teaching  in  the  larger  sense 
of  interpreting  science  in  popular  terms  for  the  public 
enlightenment, — that  they  have  been  taught  to  scorn  as 
beneath  the  dignity  of  a  doctor  (which  means  a  teacher) 
of  philosophy.  Their  estate,  I  have  said,  is  pitiable;  it 
is  pitiable;  it  is  pitiable  that  men  who  hold  themselves 
specially  trained  in  the  arts  of  scientific  "discovery"  should 
not  be  able  to  discover  the  glaringly  patent  fact  that 
research  is  often  far  easier  than  competent  exposition; 
that  every  normal  child,  for  example,  discovers  a  world 
of  facts  which  it  seldom  has  the  power  to  express  fittingly; 
and  that  little  doctors  of  philosophy  are  far  more  numer- 
ous, because  they  are  easier  to  produce,  than  great  ex- 
pounders and  interpreters  of  scientific  truth. 

When  will  scientific  specialists,  especially  those  who 
cherish  the  hope  that  the  world's  human  affairs  may  at 
length  be  scientifically  controlled  by  an  enlightened 
Democracy,  when,  I  ask,  will  such  men  keenly  feel  their 
great  obligation  to  enlighten  the  public  and  learn  to  dis- 
cipline themselves  and  their  pupils  to  keep  the  obligation? 
In  the  address  alluded  to  a  moment  ago,  I  expressed  the 
hope 

that  here  at  Columbia  or  other  competent  center  there 
may  one  day  be  established  a  magazine  that  shall  have 
for  its  aim  to  mediate  .   .   .  between  the  focal  concepts 


348  MATHEMATICAL    PHILOSOPHY 

and  the  larger  aspects  of  the  technical  doctrines  of  the 
specialist,  on  the  one  hand,  and  the  teeming  curiosity, 
the  great  listening,  waiting,  eager,  hungering  conscious- 
ness of  the  educated  thinking  public,  on  the  other. 

That  hope  has  not  yet  been  fully  realized.  But  on 
every  hand  there  are  indicia  of  amelioration:  magazines 
and  magazine  departments,  aiming  at  scientific  enlight- 
enment of  the  public,  are  growing  in  number;  scientific 
expositions  in  the  newspaper  press,  though  often  amaz- 
ingly ignorant  and  misleading,  are  becoming  less  so;  books 
of  popular  science  not  only  continue  to  multiply,  but  they 
are  better  than  formerly,  at  all  events  not  quite  so  bad; 
one  hears  more  and  more  frequently  of  universities  pro- 
viding "omnibus"  courses  in  science — general  courses, 
that  is,  designed  for  the  scientific  enlightenment  of  stu- 
dents not  intending  to  specialize  in  science,  as  one  may 
become  intelligent  about  history,  for  example,  or  music 
or  architecture,  without  becoming  a  professional  historian 
or  an  expert  musician  or  architect;  it  occasionally  happens 
that  a  university  professor,  representing  some  highly 
specialized  subject,  undertakes  to  give  instruction  in  avail- 
able parts  of  it  in  a  manner  suited  to  the  needs  of  a  gen- 
eral audience;  an  example  of  this  is  the  course  of  mathe- 
matical lectures  recently  given  in  the  University  of  Illinois 
by  Professor  J.  B.  Shaw  and  subsequently  published  in  a 
volume  entitled  The  Philosophy  of  Mathematics.  The 
signs  are  encouraging;  but  the  best  of  them  remains  to 
be  mentioned.  I  refer  to  the  recent  establishment,  at 
Washington,  of  a  new  institution,  called  Science  Service. 
The  founding  and  maintenance  of  this  institution  was 
made  possible  by  the  liberality  of  Mr.  E.  W.  Scripps,  of 
California.  Its  aim  is  that  which  I  have  been  here  trying 
to  emphasize  the  importance  of — scientific  enlightenment 


NON-EUCLIDEAN  GEOMETRIES  349 

of  the  public.  The  editor  of  Science  Service  is  Dr.  E.  E. 
Slosson.  A  good  account  by  him  of  the  new  institution's 
charter,  scope,  purpose,  organization  and  present  policy 
is  found  in  Science,  April  8,  1921.  The  work  it  is  at- 
tempting to  do  is  of  the  highest  importance  but  it  is  ex- 
ceeding hard.  It  is  hard  because  there  are  but  few  com- 
petent scientists  who  believe  in  the  possibility  of  popular- 
izing science;  because,  among  those  who  believe  in  the 
possibility,  there  are  but  few  who  are  willing  to  engage 
in  the  enterprise;  and  because,  among  the  willing,  there 
are  but  few  who  have  acquired  the  requisite  art.  What 
is  needed  is  a  more  numerous  breeding  of  men  like  Galileo 
Galilei  and  Auguste  Comte  and  W.  K.  Clifford  and 
Thomas  Huxley  and  John  Tyndall  and  Ernst  Haeckel 
and  Joseph  Le  Conte  and  Camille  Flammarion  and  Louis 
Couturat  and  Ernst  Mach  and  Josiah  Royce.  Let  the 
multiplication  of  mute  specialists  proceed — their  service 
is  mighty  in  its  way;  but  its  way  is  not  enough.  Democ- 
racy demands  that  the  discovery  of  truth  be  attended  and 
be  followed  by  exposition,  by  interpretation,  by  evalua- 
tion, in  terms  that  educated  laymen  can  understand. 

Doubtless,  you  have  been  wondering  why  these  medi- 
tations upon  the  popularization  of  science  have  been  in- 
serted as  a  prelude  to  a  lecture  on  non-Euclidean  geome- 
try. I  admit  that  they  might  have  been  submitted  else- 
where with  equal  propriety;  they  might  indeed  have  been 
presented  in  the  introductory  lecture  for  the  entire  course 
has  for  its  aim,  as  you  know,  the  democratizing  of  scien- 
tific knowledge  and  scientific  criticism.  Why,  then,  have 
I  inserted  the  discussion  in  the  present  connection?  The 
explanation,  which  is  a  very  simple  one,  is  in  terms  of 
personal  psychology.  I  had  been  considering  whether 
I  should  or  should  not  include  in  the  course  a  lecture  on 


350  MATHEMATICAL    PHILOSOPHY 

non-Euclidean  geometry,  and  what,  if  I  discussed  the 
subject  at  all,  it  would  be  best  to  say.  Well,  the  course 
of  my  meditation  respecting  the  first  question  ran  as  fol- 
lows :  the  birth  of  non-Euclidean  geometry  was  and  is  one 
of  the  most  momentous  events  in  the  history  of  thought; 
no  other  has  served  to  reveal  in  so  clear  a  light  the  nature 
of  logical  Fate  and  the  nature,  scope  and  limitation  of 
intellectual  Freedom;  no  other  has  so  well  disclosed  the 
distinction, — which  is  radical  and  cannot  be  obliterated, 
— between  the  world  of  conception  and  the  world  of  per- 
ception,— the  world  of  pure  thought  and  the  world  of 
sensuous  experience;  no  other  has  so  clearly  defined  the 
great  problem  of  ascertaining  how  the  two  worlds  are 
interrelated;  the  matter  in  question, — the  advent  and 
nature  of  non-Euclidean  geometry, — is  one  of  the  few 
great  mathematical  matters  that  professional  philosophers 
have  seriously  sought  to  understand  and  it  is  one  that 
has  at  the  same  time  persistently  haunted  the  imagination 
of  the  educated  portion  of  the  non-mathematical  public; 
many,  very  many,  have  been  the  attempts  to  explain  the 
subject  in  the  daily  press,  in  spoken  lectures,  in  magazines 
and  in  books;  nevertheless,  outside  of  mathematical 
circles,  understanding  of  the  matter,  it  must  be  owned,  is 
meagre;  for  making  the  matter  clear  to  the  man-in-the- 
street  adequate  means  has  not  yet  been  found.  What,  I 
asked  myself,  is  to  be  done?  Must  we  in  this  case  re- 
linquish the  hope  of  successful  popularization?  And  I 
answered,  no,  we  must  keep  on  trying;  for  I  vividly  re- 
called Gergonne's  noble  dream  and  the  world's  great 
hope — Democracy;  so  my  mind  was  set  swarming  with 
the  considerations  adduced  in  the  prelude;  and  that  is 
why  I  have  presented  it  here. 

Do  I  flatter  myself  with  the  belief  that  in  this  lecture 


NON-EUCLIDEAN   GEOMETRIES  351 

the  nature  of  non-Euclidean  geometry  is  at  length  going 
to  be  made  so  plain  that  he  who  runs  may  read  and  under- 
stand? I  do  not;  nothing  is  farther  beyond  my  hope. 
Much  that  others  have  said  I  shall  omit,  and  most  of 
what  I  shall  say  has  been  repeatedly  said,  in  one  way  or 
another,  by  them;  if  I  succeed  in  adding  only  a  little  light 
to  that  given  by  the  extant  literature  of  the  subject,  I 
shall  be  quite  content. 

In  Lecture  VII,  as  you  will  remember,  I  pointed  out 
that  the  term  non-Euclidean  has  two  meanings — one  of 
them  specific  and  usual,  the  other  one  generic  and  less 
usual;  the  former  meaning  always  refers  to  the  theory 
of  parallels;  the  latter  does  not.  In  the  present  discus- 
sion, the  term  will  be  used  in  the  specific  sense  only. 

It  is  customary  to  say  that  non-Euclidean  geometry 
is  a  strictly  modern  discovery,  due  to  the  daring  genius 
of  a  young  Hungarian,  John  Bolyai,  and  independently 
to  that  of  a  Russian,  Lobachevski,  both  of  whom  flour- 
ished in  the  first  half  of  last  century.  The  discovery,  as 
I  have  intimated,  was  preceded  by  an  immense  period  of 
preparation  in  which  geometricians  wrestled  with  a  very 
old  puzzle — the  so-called  problem  of  parallels.  If  you 
will  consult  Dr.  T.  L.  Heath's  superb  edition  of  Euclid's 
Elements,  you  will  find  that  controversial  discussion  of 
that  problem  began  in  pre-Euclidean  days,  was  but  ag- 
gravated instead  of  terminated  by  Euclid's  handling  of 
the  matter,  and,  though  culminating  in  the  birth  of  the 
new  geometry,  has  continued  (among  the  geometrically 
ill  informed)  down  to  our  own  day,  a  hundred  years 
after  the  puzzle  was  virtually  solved  by  the  two  pioneers 
I  have  named.  There  is  no  tale  more  romantic,  nor,  in 
the  proper  sense  of  the  term,  more  human,  in  the  whole 
history  of  Thought.     A  human  tale,   I   have  said,   dis- 


352  MATHEMATICAL   PHILOSOPHY 

tinctively  human.  I  am  unable  to  understand  how  any- 
one can  ponder  its  character  and  its  significance  intelli- 
gently and  candidly  without  seeing  clearly  that  the  old 
zoological  conception  according  to  which  human  beings 
are  a  species  of  animal  is  not  only  false,  as  Count  Korzyb- 
ski  has  pointed  out,  but  is  stupid  as  well;  in  view  of  its 
baneful  effects  upon  the  world's  ethics,  the  monstrous 
misconception  deserves  indeed  to  be  branded  as  the  Great 
Stupidity. 

I  shall  not  here  recount  the  tale;  the  story  has  been 
often  told  in  all  the  lands  and  all  the  tongues  of  science. 
If  you  desire  to  learn  the  story,  you  will  find  in  the  men- 
tioned work  of  Dr.  Heath  an  ample  clue  to  the  litera- 
ture. For  an  account  that  is  at  once  very  clear,  very 
succinct  and  finely  critical,  I  have  special  pleasure  in  re- 
ferring to  Dr.  George  B.  Halsted's  article  ("Geometry, 
non-Euclidean")  in  the  Encyclopedia  Americana.  I  own 
to  the  feeling  of  a  little  pride, — pardonable,  I  hope, — 
in  that  citation  for  it  was  my  privilege,  as  then  mathe- 
matical editor  of  Americana,  and  my  good  fortune,  to 
obtain  the  article  in  question.  I  requested  Dr.  Halsted 
to  write  it  because  he  was  specially  qualified  to  do  it; 
no  other  American  scholar  knew  more  than  he  of  non- 
Euclidean  origins  and  no  other  has  done  so  much  as  he 
has  done,  by  voice  and  pen,  to  signalize  the  importance 
of  non-Euclidean  geometry  as  making  and  marking  a 
momentous  epoch  in  human  Thought. 

Of  non-Euclidean  geometry  there  are  two  principal 
varieties;  these  are  associated  respectively  with  the  names 
of  their  inventors — Lobachevski  (1793-1856)  and  Bern- 
hard  Riemann  (1826-66).  I  am  going  to  tell,  as  clearly 
as  I  can  without  too  great  prolixity,  how  the  varieties 


NON-EUCLIDEAN  GEOMETRIES  353 

arise  and  what  they  are.     And  I  will  begin  with  that  of 
the  Russian. 

The  point  of  departure  is  Euclid's  famous  postulate, 
— his  postulate  V, — which  I  stated  in  a  previous  Lecture 
(VII)  ;  this  postulate, — or  assumption,  for  that  is  what 
it  is, — is  pretty  long;  it  is  known,  however,  to  be  equiva- 
lent to  the  briefer  assumption: 

Through  any  point  there  is  one  and  but  one  line, 
parallel  to  a  given  line. 

For  many  centuries  geometricians,  great  and  small, 
tried  to  deduce  this  assumption, — which  may  be  called 
the  ow^-parallel  assumption, — as  a  theorem  from  Euclid's 
other  assumptions  (conscious  and  unconscious).  They 
failed,  and  today  we  know  why — the  assumption  is  not 
implied  in  the  other  ones  and  so  is  not  deducible  from 
them,  not  even  by  demons  or  archangels  or  gods.  Now, 
what  the  adventurous  spirit  of  Lobachevski  led  him  to 
do  is  simply  this:  retaining  all  of  the  Euclidean  assump- 
tions save  the  one  respecting  parallels,  he  replaced  the 
latter  by  an  assumption  contradicting  it,  and  then  pro- 
ceeded to  deduce  the  consequences  of  the  set  of  assump- 
tions he  had  thus  adopted  as  postulates.  What  is  the 
assumption  with  which  he  replaced  Euclid's  postulate  of 
the  single  parallel?  It  may  be  stated  as  follows  by  help 
of  Fig.  30:  //  line  PF  rotate  (in  the  plane  of  the  figure) 
about  P,  say  counterclockwise ,  it  will  come  to  a  position, 
call  it  PK,  where  it  first  fails  to  cut  line  L,  and  then,  with- 
out cutting  L,  it  will  rotate  through  a  finite  angle  KPH 
into  a  position  PH  such  that,  if  it  rotate  further,  it  will 
cut  L  to  the  left  of  F,  the  angles  FPK  and  FPK'  being 
equal. 

According  to  this  assumption  the  lines  of  the  Pencil 


354 


MATHEMATICAL   PHILOSOPHY 


P  fall  into  two  sets — the  set  of  those  that  cut  L  and  the 
set  of  those  that  do  not;  the  former  set  consists  of  all 
the  lines  within  the  angle  KPK! ;  and  the  latter  set  con- 
sists of  PK,  PK'  and  the  lines  within  the  angle  KPH  (or, 
what  is  tantamount,  the  angle  K'PH').  The  limiting  or 
boundary  non-intersectors,  PK  and  PK' ,  are  the  Lo- 
bachevski  parallels — parallel,  that  is,  to  L.  We  may, 
accordingly,  call  Lobachevski's  asumption  the  /wo-parallel 
assumption. 


$^ 

"\\ 

9Q° 

^^H 

\K 

/k 

F 

A\ 

L 

Fig.  30. 

There  remains  a  third  possibility,  pointed  out  in  1854 
by  Riemann  in  his  famous  inaugural  dissertation,  with 
which  professional  philosophers  seem  to  have  but  little 
or  no  acquaintance,  though  it  involves  the  complete 
demolition  of  the  Kantian  conception  of  space  and  though 
its  thought  is  sufficient  to  have  immortalized  several  men 
had  they  severally  originated  its  several  parts.  I  refer  to 
the  lecture,  On  the  Hypotheses  which  lie  at  the  Basis  of 
Geometry.  Translated  into  English  by  W.  K.  Clifford, 
it  is  found  in  the  latter's  Collected  Works.     In  the  ad- 


NON-EUCLIDEAN  GEOMETRIES  355 

dress  Riemann  indicated  the  possibility  of  constructing 
a  geometry  upon  a  basis  of  postulates  containing  the  as- 
sumption that 

No  two  lines  of  a  plane  are  parallel;  or,  what  is 
equivalent,  every  two  lines  of  a  plane  intersect. 

This  Mo-parallel  assumption  contradicts  not  only  the 
OHtf-parallel  assumption  of  Euclid  but  another  assumption 
of  his, — a  tacit  assumption, — namely,  that  a  line  has  in- 
finite extent  (see  propositions  16  and  28,  Book  I,  in 
Heath's  edition  of  the  Elements).  The  remaining  as- 
sumptions of  Euclid  are  retained  by  Riemann. 

We  have  now  before  us  three  postulate  systems, — 
one  of  them  Greek, — one  of  them  Russian, — one  of  them 
German.  Upon  them  have  been  erected  and  now  stand 
three  geometries,  one  of  them  Euclidean,  the  other  two 
non-Euclidean;  of  these  two,  the  former  is  often  called 
Lobachevskian,  and  the  latter  Riemannian.  For  a  good 
reason,  which  I  will  not  pause  here  to  explain,  Professor 
Felix  Klein  has  called  the  three  geometries  respectively 
Parabolic,  Hyperbolic,  and  Elliptic,  descriptions  that  are 
now  in  common  use. 

With  the  elements  of  the  parabolic  geometry  you  are 
familiar;  with  those  of  the  other  two  varieties  you  are 
presumably  not  acquainted.  The  hyperbolic  and  elliptic 
geometries  have  been  built  up  by  various  methods,  ele- 
mentary and  more  advanced,  pure  and  analytic.  I  credit 
you  with  having  curiosity  to  see  how  the  building  can  be 
done  by  the  familiar  elementary  processes  of  ordinary 
geometry.  To  apply  them  here  would  detain  us  too  long; 
but  if  you  have  the  curiosity,  you  can  gratify  it  by  reading 
the  previously  mentioned  essay  of  Professor  F.  S.  Woods 
on  "Non-Euclidean  Geometry"    (found  in  Monographs 


356  MATHEMATICAL   PHILOSOPHY 

on  Modern  Mathematics,  edited  by  J.  W.  A.  Young). 

We  have  seen  in  what  respects  the  bases,  the  postulate 
systems,  of  the  three  geometries  are  alike  and  in  what 
respects  they  are  unlike.  We  naturally  pass  to  a  com- 
parison of  the  superstructures — to  a  comparison,  that  is, 
of  the  theorematic  contents  of  the  geometries.  Do  the 
geometries  intersect?  Have  they,  that  is,  any  theorems 
in  common?  The  answer  is  obvious:  they  have  in  com- 
mon such  and  only  such  theorems  as  are  deducible  from 
the  assumptions  that  are  common  to  the  three  systems 
thereof.  One  such  theorem  is  this:  The  summit  angles 
of  a  birectangular  quadrilateral  are  equal;  in  other  words, 
if  ABCD  be  a  quadrilateral  having  right  angles  at  A  and 
B  and  having  the  side  AC  equal  to  the  side  BD,  then  the 
angle  at  C  is  equal  to  that  at  D.  You  may  wish  to  make 
a  list  of  such  common  theorems  as  an  exercise.  More 
striking  are  the  theorems  in  which  the  geometries  differ; 
such  differences  are  of  course  due  to  the  differences  in  the 
postulate  systems.  Let  us  notice  some  of  them.  The 
one  most  commonly  mentioned  relates  to  the  sum  of  the 
angles  of  a  triangle.  In  the  parabolic  geometry  that  sum 
is  constant  (the  same  for  all  triangles)  and  is  exactly  two 
right  angles,  as  you  know;  in  the  hyperbolic  and  elliptic 
geometries  the  sum  is  variable  (depending  upon  the  tri- 
angle's size)  ;  in  the  former  geometry  it  is  always  less 
than  two  right  angles  and  decreases  as  the  triangle's  area 
increases;  in  the  latter,  the  sum  is  always  greater  than 
two  right  angles  and  increases  with  the  area. 

Again :  If  two  lines  of  a  plane  are  perpendicular  to  a 
third,  then,  in  parabolic  geometry,  the  two  are  parallel; 
in  the  hyperbolic,  they  are  not  parallel  nor  do  they  inter- 
sect; in  the  elliptic,  they  meet  at  a  point  whose  distance 


NON-EUCLIDEAN  GEOMETRIES  357 

from  the  third  line  is  finite,  and  all  perpendiculars  to  this 
line  meet  at  that  point. 

One  more :  In  parabolic  geometry  each  summit  angle 
of  an  isosceles  birectangular  quadrilateral  is  a  right  angle; 
in  hyperbolic  geometry,  it  is  acute;  and  in  elliptic  geome- 
try it  is  obtuse.  So  it  is  seen  that  neither  of  the  non- 
Euclidean  geometries  contains  rectangles  among  its 
figures. 

It  would  be  easy  to  prolong  the  list  of  such  differ- 
ences; the  theorems  already  stated,  which  are  not  dif- 
ficult to  prove,  are  sufficient,  however,  for  illustration  and 
they  ought,  I  think,  to  challenge  the  curiosity  of  any  in- 
tellectual student. 

There  are  certain  questions  which  you  are  doubtless 
bursting  to  ask,  for  in  a  discussion  of  this  subject  thought- 
ful beginners  always  ask  them. 

One  of  the  questions  is  this:  Can  we  be  quite  cer- 
tain that  neither  of  the  non-Euclidean  geometries  involves 
an  inner  contradiction?  In  other  words,  can  we  be  cer- 
tain with  respect  to  each  of  them  that  the  propositions 
constituting  it  are  compatible  with  one  another?  The 
answer  is,  yes.  The  propositions  constituting  a  geometry 
consist,  as  you  know,  of  its  postulates  and  of  the  proposi- 
tions logically  deducible  therefrom;  and  so,  if  the  postu- 
lates are  mutually  compatible,  the  whole  geometry  is  self- 
consistent.  The  question  thus  reduces  to  this:  Can  we 
be  certain  with  respect  to  each  of  the  non-Euclidean 
geometries  that  its  postulates  are  mutually  compatible? 
Now,  in  respect  of  the  Euclidean  postulates,  we  saw  in 
Lecture  VI,  you  will  remember,  that  we  can  be  as  cer- 
tain of  their  compatibility  as  it  is  possible  to  be  of  any 
reasoned  proposition,  and  that  is  what  I  mean  by  "quite 
certain"  and  it  is  what  you  mean.     Well,  it  can  be  shown 


858  MATHEMATICAL   PHILOSOPHY 

that,  if  Euclidean  geometry  is  self-consistent,  then  each  of 
the  non-Euclidean  geometries  is  self-consistent.  And  this 
has  been  done  by  various  mathematicians  in  various  ways. 
It  has  been  done  very  simply  by  Henri  Poincare  in  his 
widely  known  'Science  and  Hypothesis;  and  it  has  been 
done  still  better,  in  the  sense  of  greater  detail,  in  the  pre- 
viously cited  Elementare  Geometrie  of  Weber  and  Well- 
stein.  Suffice  it  here  to  say, — for  I  am  going  to  leave  it 
to  you  to  examine  the  proofs, — that  the  principle  or  the 
trick  involved  in  them  is  that  of  showing  the  postulate 
systems  of  the  non-Euclidean  geometries  to  be  each  of 
them  satisfied  by  suitably  selected  classes  of  geometric 
entities  found  in  Euclidean  geometry.  So,  you  see,  if 
the  non-Euclidean  geometries  have  any  unsoundness  in 
them,  there  is  a  corresponding  unsoundness  in  Euclidean 
geometry.  In  respect  of  soundness — inner  consistency — 
self-compatibility — logical  concordance  among  the  parts 
of  each — the  three  geometries  are  on  exactly  the  same 
level,  and  the  level  is  the  highest  that  man  has  attained. 
The  three  doctrines  are  equally  legitimate  children  of  one 
spirit, — the  geometrizing  spirit,  which  Plato  thought 
divine, — and  they  are  immortal.  Work  inspired  and  ap- 
proved by  the  muse  of  intellectual  harmony  can  not  per- 
ish— it  is  everlasting. 

Another  question  is:  Are  these  geometries  true? 
They  are  true  in  the  sense  in  which  truth  resides  in  a 
body  of  propositions  of  which  some  are  mutually  com- 
patible premises  and  the  rest  are  inevitable  consequences 
thereof,  enchained  thereto  by  the  binding  threads  of  log- 
ical fate,  which  is  changeless  and  timeless.  Such  truth, 
however,  though  it  is  ineffably  precious,  is  only  a  quality 
or  an  aspect  of  that  inner  consistency  which  gives  an  au- 
tonomous body  of  propositions  its  peculiar  beauty,  pure, 


NON-EUCLIDEAN  GEOMETRIES  359 

perfect,  and  eternal.  But  it  is  not  this  aspect  of  logical 
consistency  that  the  "true"  of  your  question  is  designed 
to  mean. 

Perhaps  the  question  you  intend  to  ask  is  this:  Are 
the  geometries  true  in  the  sense  of  giving  an  exact  ac- 
count of  Space?  Or,  better,  since  they  contradict  one 
another  in  cardinal  matters,  is  one  of  them  true  in  the 
indicated  sense?  I  have  already  pointed  out  that  the 
term  "space"  does  not  occur  in  Euclid's  Elements,  and 
I  may  add  that  there  is  no  necessity  for  its  occurrence  in 
either  of  the  non-Euclidean  geometries.  Since  it  is  never- 
theless customary  to  use  the  term  in  philosophic  discus- 
sions of  geometry,  we,  too,  must  do  so  here.  If  we  are 
to  do  so  profitably,  we  must  make  and  keep  steadily  in 
mind  a  fundamental  distinction,  which  is  indeed  a  pretty 
obvious  one  but  is  commonly  neglected;  and  the  neglect 
of  it  is  always  attended  with  utter  confusion.  We  must, 
I  mean,  not  fail  to  distinguish  sharply  between  perceptual 
space  and  spaces  of  conception;  that  is,  between  space  in 
which  points,  lines,  planes,  circles,  spheres,  and  so  on,  are 
material  or  physical  dots,  rods,  or  ropes,  slabs  or  rough 
irregular  "surfaces"  thereof,  hoops  or  rings,  globes  or 
balls  (of  wood  or  gold  or  marble),  and  so  on,  and  a 
space  in  which  the  terms  point,  line,  .  .  .  denote  pure 
concepts  of  which  no  instance  is  found,  for  no  instance 
exists,  in  perceptual  space.  Unless  we  make  and  keep 
that  radical  distinction,  we  might  better  abandon  the 
discussion;  but  if  we  make  the  distinction  clearly  and  do 
not  lose  it,  the  question  you  have  put  can  be  answered 
clearly  and  rightly. 

And  the  answer?     It  is  found  in  the  following  con- 
siderations. 

The  space  of  a  geometry  is  always  a  conceptual  space. 


360  MATHEMATICAL   PHILOSOPHY 

What  is  the  space?  It  is  the  class  of  conceptual  entities 
(no  matter  what  we  call  them,  points,  lines,  and  so  on, 
or  jabberwockies)  which  satisfy  or  verify  the  geometry's 
postulates  and  about  which  the  geometry  is  therefore  a 
reasoned  discourse.  The  geometry  is,  therefore,  true  in 
the  sense  of  giving  an  exact  account  of  space,  where  by 
"space"  is  meant  the  space  of  that  geometry.  I  say  "of 
that  geometry"  for,  you  see,  two  geometries  which  con- 
tradict one  another  in  one  or  more  respects  have  dif- 
ferent spaces.  The  answer  to  your  question  is,  then,  this: 
Euclidean  geometry,  Lobachevskian  geometry,  and  Rie- 
mannian  geometry  are  each  of  them  true  in  the  sense  that 
each  of  them  gives  an  exact  account  of  its  own  space, 
which  is  a  conceptual  space. 

But  what  of  perceptual  space?  What,  I  mean,  of 
that  all-enveloping  region  or  room  or  spread  which  is 
revealed  to  us  by  touch  and  sight  and  hearing  and  the 
sense  of  muscular  movement?  Is  one  of  the  three  geome- 
tries true  in  the  sense  of  giving  an  exact  account  of  this 
space?  The  question  is  a  fallacy  of  interrogation;  it  im- 
plies, that  is,  that  our  perceptual  space  is  a  thing  of  which 
an  exact  geometric  account  is  possible;  but  it  is  not  such 
a  thing — perceptual  space  is  not,  rightly  speaking,  ge- 
ometrizable.  Wherein  it  fails  to  be  so,  is  easy  to  make 
clear.  Consider,  for  example,  three  of  its  "lines,"  say 
pencils  or  rods,  lu  /2,  /3.  Compare  their  "lengths"  per- 
ceptually, which  is  the  only  way  in  which  such  "lengths" 
can  be  compared.  Compare  h  with  /.,  then  l2  with  4, 
and  then  h  with  /-,.  You  know  what  may  happen,  for  it 
is  a  common  phenomenon  of  such  comparison  of  per- 
ceptual things.  It  may  happen  that  the  "length"  of  h 
is  equal  to  (indistinguishable  from)  the  "length"  of  /2, 
that  the   "length"   of  /2  is  equal  to  the   "length"   of  lS} 


NON-EUCLIDEAN  GEOMETRIES  361 

while  that  of  lx  is  z/wequal  to  (distinguishable  from)  that 
of  /3,  so  that,  in  symbols,  the  record  of  relations  will 
stand  thus: 

(i)     Ji=/2,    /2=4,    li9*h. 

Now  reflect  that  our  perceptual  space  is  such  that  the 
the  concurrence  of  relations  like  (i)  is  a  familiar  and 
unavoidable  phenomenon  of  "lengths"  not  only  but  also 
of  "areas"  and  "volumes";  then  reflect  that  such  situa- 
tions as  ( i ) , — though  inherent  in  the  nature  of  percep- 
tual space, — are  utterly  and  glaringly  illogical;  finally, 
reflect  that  all  the  relations  occurring  in  a  geometry  must 
be,  unlike  those  of  (i),  logically  consistent;  and  you  will 
be  thus  led  to  see  clearly  that  our  perceptual  space  cannot 
be  geometrized  in  Euclidean  fashion  or  otherwise,  if  the 
term  geometry  is  to  retain  its  essential  meaning. 

But  is  there  not,  among  the  various  current  meanings 
of  the  term  "truth,"  one  meaning  which  enables  us  to  say 
that  Euclidean  geometry,  regarded  as  a  doctrine  about 
our  perceptual  space,  is  true?  The  answer  is:  yes,  there 
is  such  a  meaning.  It  is  the  "instrumental"  meaning  in- 
sisted upon  by  Professor  John  Dewey — the  "pragmatic" 
meaning  first  signalized  by  C.  S.  Peirce,  subsequently  in- 
terpreted, elaborated  and  advocated  by  William  James 
and  others.  It  is  the  meaning  in  accordance  with  which 
an  idea  or  a  proposition  or  a  doctrine  is  true  if  it  "works," 
in  so  far  as  it  "works,"  so  long  as  it  "works."  The 
meaning  is  not  without  merits  that  commend  it  to  all 
men  and  women  for  all  human  beings  have,  below  their 
distinctively  human  qualities,  certain  animal  propensities 
and  animal  impulses,  and,  in  the  animal  world,  the  end 
always  justifies  the  means — all  ways  that  "work"  equally 
well,  all  means  that  are  equally  "effective,"  are  equally 


362  MATHEMATICAL   PHILOSOPHY 

good.  It  is  the  pragmatic  meaning  of  truth  that  makes 
treason  a  crime,  if  it  fail,  and  a  virtue,  if  it  succeed.  It 
is  a  meaning  that  is  especially  congenial  to  practicians 
and  "politicians,"  whose  "philosophy"  never  rises  above 
the  question:  How  can  I  "get  there"?  How  can  I  put 
this  thing  "through,"  "over"  or  "across"?  What  is  the 
means  that  will  "work"?  It  is  a  meaning,  too,  that  is 
especially  congenial  to  an  industrial  age, — an  experi- 
mental age, — an  age  of  laboratories, — an  adventurous 
age  when  men  act  more  than  they  think.  In  the  head- 
long rush  and  hurly-burly  of  such  an  age,  men  and  women 
are  not  aware  of  the  fact  that  the  world  of  human  affairs 
would  quickly  dash  upon  utter  destruction  but  for  the 
guiding  and  saving  influence  of  a  nobler  truth-conception 
which  they  do  not  consciously  own, — the  conception,  I 
mean,  of  truth  as  having  its  highest  meaning  in  the  un- 
changing relations  and  eternal  laws  of  Logical  Thought. 

I  have  said  that  in  the  pragmatic  sense  of  "true," 
Euclidean  geometry  may  be  said  to  be  true  of  our  per- 
ceptual space.  It  may  be  said  because  this  geometry, 
when  applied  (as  we  say)  to  this  space,  "works" — which 
means  that  temples,  aqueducts,  tunnels,  bridges,  railways, 
ships  and  other  architectural  and  engineering  structures 
whose  designing  is  guided  by  Euclidean  formulas  are 
successful — they  wear  out  but  they  do  not  perish  from 
any  essential  defect  of  structural  design. 

Does  not  the  fact  just  stated  show  that  Euclidean 
geometry  is  superior  to  the  other  two  varieties?  It  does 
not;  for,  if  the  designing  and  the  building  of  the  struc- 
tures alluded  to  were  guided  by  Lobachevskian  or  by 
Riemannian  formulas,  the  structures,  when  completed, 
would  not  differ  perceptibly  from  the  former  ones,  and 
the  railways,  ships,  and  so  on,  would  be  equally  durable 


NON-EUCLIDEAN  GEOMETRIES  363 

and  serviceable;  the  reason  is  that,  though  the  formulas 
of  any  one  of  the  geometries  differ  radically  from  their 
correspondents  in  either  of  the  other  two,  yet  they  differ 
in  such  a  way  that  the  difference  could  not  crop  out  in 
any  physical  structure  unless  the  latter  were  vastly  larger 
than  any  our  small  planet  admits  of.  We  may  say, 
then,  that  all  three  of  the  geometries  are  pragmatically, 
or  instrumentally,  true  of  our  perceptual  space — the 
space  of  sensuous  experience,  though,  as  we  have  seen, 
no  one  of  them  and  no  other  geometry  is  or  can  be, 
rightly  speaking,  the  geometry  or  a  geometry  of  percep- 
tual space. 

Since  all  three  of  the  geometries  in  question  are  prag- 
matically true  of  our  perceptual  space,  why  is  it  that  in 
practical  work,  like  bridge  building  and  the  like,  the 
Euclidean  variety  is  employed  exclusively?  It  is  because 
the  Euclidean  formulas  are  simpler,  easier  to  use  than 
the  others.  What,  if  any,  is  the  epistemological  signifi- 
cance of  this  fact?  The  question  seems  important.  I  do 
not  know  the  answer.    Maybe  one  of  you  will  discover  it. 

Is  non-Euclidean  geometry  always  3-dimensional? 
No;  like  the  Euclidean  variety,  it  may  have  any  number 
of  dimensions  from  one  up. 

What  of  Einstein  geometry?  The  answer  is  implicit 
in  what  has  been  said.  "Einstein  geometry"  is  not  geome- 
try,— not  yet,  at  all  events, — it  is  a  figure  of  speech,  con- 
venient for  experts,  misleading  for  laymen.  That  is  not 
a  comment  upon  the  doctrine  of  Relativity  regarded  as 
being, — what  it  is, — a  physical  theory. 

The  advent  of  non-Euclidean  geometry  is,  I  have  said, 
one  of  the  gravest  events  in  the  history  of  thought.  It 
has  been  tragic  as  well.  The  two  facts  are  connected. 
Thirty  years  ago,  I  visited  a  locally  eminent  professor  of 


364  MATHEMATICAL   PHILOSOPHY 

mathematics  in  an  excellent  middle-West  college  of  the 
sectarian  variety.  I  was  astonished  to  find  him  in  a  sad 
mental  state,  worried,  distracted,  agitated,  tremulous, 
unable  to  sleep  or  rest,  thinking  always  about  the  same 
thing,  and  no  longer  able  to  do  so  coherently.  What 
was  the  trouble?  For  many  years  he  had  been  teaching 
geometry, — Euclidean  geometry, — and  his  teaching  had 
been  done  in  the  spirit  and  faith  of  a  venerable  philoso- 
phy. Like  almost  all  the  educated  men  of  his  time  and 
like  millions  of  others  in  the  preceding  centuries,  he  had 
been  bred  in  the  belief  that  the  geometry  he  was  teach- 
ing was  far  more  than  a  body  of  logical  compatibilities; 
it  was  not  only  true  internally, — logically  sound,  that  is, 
— but  it  was  true  externally — an  exact  account  of  space, 
the  space  of  the  sky  and  the  stars;  its  axioms  were  not 
mere  assumptions, — not  mere  ifs, — they  were  truths, 
"self-evident"  truths,  and,  like  the  propositions  implied 
by  them,  they  were  not  only  valid  but  were  known  to  be 
valid,  and  valid  eternally;  in  a  word,  the  geometry  of 
Euclid  was  a  body  of  absolute  knowledge  of  the  nature 
of  space, — the  space  of  the  outer  world, — other  space 
there  was  none.  That  was  a  comforting  belief,  a  con- 
genial philosophy,  held  as  a  precious  support  of  religion 
and  life;  for,  though  there  are  many  things  unknown  and 
some  perhaps  unknowable,  yet  something,  you  see,  was 
known;  there  was  thus  a  limit  to  rational  skepticism;  our 
human  longing  for  certitude  had  at  least  one  great  grat- 
ification— the  validity  of  Euclidean  geometry  as  a  de- 
scription of  Space  was  indubitable.  Such  was  the  philoso- 
phy in  which  my  dear  old  friend  had  been  bred,  and, 
with  unquestioning  confidence,  he  had  devoted  long  years 
to  the  breeding  of  others  in  it.  At  length,  he  heard  of 
non-Euclidean  geometries,  in  which  his  cherished  certi- 


NON-EUCLIDEAN  GEOMETRIES  365 

tudes  were  denied — denied,  he  knew,  by  great  mathe- 
maticians, by  men  of  creative  genius  of  the  highest  order; 
he  could  not  accept,  he  could  not  reject,  he  could  not 
reconcile;  the  foundations  of  rational  life  seemed  utterly 
destroyed;  he  pondered  and  pondered  but  the  great  new 
meaning  he  was  too  old  to  grasp,  and  his  mind  perished 
in  the  attempt, — killed  by  the  advancement  of  science, — 
slain  by  a  revolution  of  thought. 

Pitiless  indeed  are  the  processes  of  Time  and  Crea- 
tive Thought  and  Logic;  they  respect  the  convenience  of 
none  nor  the  love  of  things  held  sacred;  agony  attends 
their  course.  Yet  their  work  is  the  increasing  glory  of 
a  world, — the  production  of  psychic  light, — the  growth 
of  knowledge, — the  advancement  of  understanding, — the 
enlargement  of  human  life, — the  emancipation  of  Man. 


LECTURE  XVIII 
The  Mathematics  of  Psychology 

BACKWARDNESS   OF   THIS   AND   OF   THE   PSYCHOLOGY   OF 

MATHEMATICS THE  LAW  OF  WEBER  AND  FECHNER 

REEXAMINED SOME  OF  THE  LAW'S  UNNOTICED  IM- 
PLICATIONS  THOUGHT     AS     INFINITELY     REFINED 

SENSIBILITY. 

Two  of  the  subjects  that  this  course  should  include 
are  the  mathematics  of  psychology,  and  the  psychology 
of  mathematics.  The  two  things,  though  closely  related, 
are  distinct — they  interpenetrate  but  neither  includes  the 
other;  it  is  one  thing  to  mathematicize  psychology,  and 
a  very  different  thing  to  psychologize  mathematics;  the 
aim  of  the  former  is  to  express  psychological  relations  in 
mathematical  terms;  that  of  the  latter  is  to  study  those 
aspects  of  mathematics  which  are  psychological  as  dis- 
tinguished from  logical;  both  enterprises  are  immense 
and  important;  neither  one  is  far  advanced,  and  the  rea- 
sons are  evident — the  tasks  are  difficult,  requiring  special 
preparation,  while  psychologists  have  not  been  mathe- 
maticians and  the  latter  have  not  been  psychologists.  But 
psychologists  have  done  as  much  to  mathematicize  their 
subject  as  mathematicians  have  done  to  understand  the 
psychology  of  theirs,  though  the  former  task  is  perhaps 
the  harder  of  the  two. 

366 


THE    MATHEMATICS    OF    PSYCHOLOGY      367 

I  will  begin  with  the  mathematics  of  psychology  and 
will  in  the  main  confine  my  remarks  to  its  most  famous 
achievement,  which  is  also,  I  believe,  its  most  important 
one — the  so-called  Psychophysical  Law  of  Weber  and 
Fechner.  When  the  law  was  duly  announced  in  its  mathe- 
matical form,  it  aroused  much  interest  among  psycholo- 
gists, evoked  much  admiration,  and  stimulated  research 
and  discussion;  at  the  same  time  it  produced  something 
like  fright  or  consternation  for  it  began  to  seem  that  one 
could  not  be  a  scientific  psychologist  without  being  a 
mathematician,  and  that  was  a  fearful  thought.  But  the 
"interest,"  the  "admiration"  and  the  "fright"  were  des- 
tined to  pass  or,  at  least,  to  suffer  much  mitigation.  A 
generation  ago  the  law  had  been  often  presented  and 
elaborately  discussed.  At  length  it  was  handled  by  Will- 
iam James  in  The  Principles  of  Psychology.  After  pre- 
senting it  with  characteristic  honesty  and  with  quite  as 
much  accuracy  as  could  be  expected  from  one  who  not 
only  was  not  a  mathematician  but  knew  and  owned  that 
he  was  not,  James  proceeds  to  examine  the  claims  that 
had  been  made  in  behalf  of  the  law  and  then,  with  abso- 
lute candor  and  great  confidence,  to  estimate  its  signifi- 
cance. And  what  is  the  estimate?  It  is  this:  "Fechner's 
book,  Psychophysik,  was  the  starting  point  of  a  new  de- 
partment of  literature,  which  it  would  be  perhaps  impos- 
sible to  match  for  the  qualities  of  thoroughness  and  sub- 
tlety, but  of  which,  in  the  humble  opinion  of  the  present 
writer,  the  proper  psychological  outcome  is  just  nothing" 
(Vol.  I,  p.  534).  Again:  "The  Fechnerian  Maasformel 
and  the  conception  of  it  as  an  ultimate  'psychophysic  law' 
will  remain  an  Mdol  of  the  den,'  if  there  ever  was  one" 
(P-  549)  •  Of  that  judgment,  right  or  wrong,  pronounced 
by  so  great  an  authority,  it  may  not  be  said  that  its  psycho- 


368  MATHEMATICAL   PHILOSOPHY 

logical  "outcome"  or  effect — upon  psychologists — was 
"just  nothing."  On  the  contrary  the  effect  was  great  and 
it  was  soothing;  it  dampened  interest  in  Fechner's  work; 
it  moderated  admiration  of  the  man;  and  it  greatly  re- 
lieved many  a  psychologist  who  had  been  frightened  by 
what  had  seemed  a  serious  mathematical  invasion  of  his 
subject.  James's  discussion  of  the  matter  is  very  inter- 
esting and  enlightening — more  so  than  any  other  I  have 
seen;  his  presentation  of  the  law  in  question  is  marred, 
however,  by  some  inaccuracies;  moreover,  respecting  the 
significance  of  the  law,  it  has,  if  I  be  not  mistaken,  certain 
implications  and  important  bearings  not  noticed  by  James 
nor,  I  believe,  by  others.  It  has,  therefore,  seemed  to 
me  that  a  discussion  of  the  matter  might  be  properly  in- 
cluded in  this  course,  even  though  part  of  my  remarks 
can  at  most  remind  you  of  things  you  are  already  fa- 
miliar with. 

The  Rise  of  Psychology  as  an  Experimental  Science. 
— For  convenience  of  reference  let  me  place  before  you, 
in  chronological  order,  five  important  names. 

Immanuel  Kant  (i 724-1 804) 
Johann  Friedrich  Herbart  (1776-1841) 
Ernst  Heinrich  Weber  ( 1 796-1 878 ) 
Gustav  Theodor  Fechner  (1801-1887) 
Bernhard  Riemann   (1 826-1 866) 

I  well  remember  that,  when  I  was  a  boy,  it  was  cus- 
tomary for  people  who  liked  to  talk  about  science  to 
speak  of  two  kinds  thereof — the  natural,  or  physical, 
sciences  and  the  so-called  mental  sciences.  The  classifica- 
tion was  then  an  old  one  but  it  is  not  yet  without  some 
vogue.  The  distinction  may  not  have  been  profound  but 
it  was  obvious:  the  former  kind  of  science  was  quantita- 


THE    MATHEMATICS    OF    PSYCHOLOGY      309 

tive  for  it  dealt  with  phenomena  that  were  measurable; 
the  latter  kind  was  qualitative  for  it  dealt  with  phe- 
nomena that  were  supposed  to  be  non-measurable.  Now, 
owing  to  its  lack  of  precision,  a  qualitative  science  was 
looked  upon  as  being  scientifically  inferior  to  one  that  was 
quantitative.  According  to  Kant,  for  example,  psy- 
chology was  not  a  true  science  and  never  could  be.  What 
he  meant  by  a  "true  science"  is  sufficiently  revealed  by 
his  saying  that  a  natural  science  is  a  science  only  in  so 
far  as  it  is  mathematical.  It  is  noteworthy  that  this  say- 
ing of  a  great  philosopher  accords  perfectly  with  the 
saying  of  one  who  was  at  once  a  great  physicist  and  a 
great  mathematician — Bernhard  Riemann:  "Natural 
Science  is  the  attempt  to  understand  nature  by  means  of 
exact  concepts."  It  is  "noteworthy"  but  is  not  surpris- 
ing in  view  of  Kant's  strong  predilection  for  mathematics 
and  physical  science,  and  of  Riemann's  early  interest  in 
psychology  and  metaphysics  (as  shown  in  the  Anhang  of 
his  Gesammelte  Werke  or  in  Keyser's  translation  of  the 
Anhang  in  the  Monist,  January,  1900). 

Among  the  first  to  reject  the  Kantian  dogma  respect- 
ing psychology  was  Herbart,  the  so-called  "exact  philoso- 
pher." Herbart  believed  it  was  possible  to  build  up  a 
static:,  and  a  mechanics  of  elementary  ideas, — a  mechanics 
of  mind,  let  us  say, — modeled  after  the  classical  me- 
chanics of  matter,  and  he  boldly  essayed  the  task.  But 
apriori  reasoning  cannot  do  a  work  that  calls  for  patient 
experimentation,  and  Herbart's  mathematical  formulae, 
though  rather  impressive  to  the  physical  eye,  have  but 
little  interest  except  as  representing  a  reaction  and  a 
prophecy. 

It  was  not  a  professional  philosopher  who  took  the 
first  important  steps  toward  making  psychology  a  labora- 


370  MATHEMATICAL   PHILOSOPHY 

tory,  or  quantitative,  science;  they  were  taken  by  men 
trained  in  the  ways  of  natural  science.  These  men  were 
Weber,  professor  of  anatomy  and  physiology  in  the  uni- 
versity of  Leipzig,  and  Fechner,  professor  of  physics  in 
the  same  institution.  What  was  the  new  problem  they 
set  for  themselves  and  how  did  they  attack  it?  Well, 
there  is  in  our  world  what  we  call  matter  and  there  is 
what  we  call  mind.  Let  us  not  tarry  to  debate  the  great 
present-day  question  whether  the  two  things  are  essen- 
tially one  nor  whether  they  are  derived  from  a  "neutral" 
something — something,  that  is,  that  is  neither  matter  nor 
mind.1  For  our  pioneers  mind  and  matter  were  obviously 
two,  the  two  were  related,  and  the  problem  was  to  ascer- 
tain how.  And  their  method  was  that  of  experiment  and 
observation.    Where  did  they  begin?    And  why  there? 

Sense  Departments  and  Their  Fundamental  Prob- 
lems.— Imagination,  conception,  reason,  will,  all  these 
have  to  do  with  matter,  but  experimental  research  did  not 
begin  with  them.  Why  not?  Because  it  was  best  to  be- 
gin at  the  beginning,  and  the  beginning  is  sensation — it  is 
in  what  we  call  sensation  that  our  "minds"  first  get  into 
some  sort  of  knowing  connection  with  "matter."  It  was 
soon  found  necessary  to  distinguish  many  more  "sense 
departments," — as  they  are  called, — than  the  traditional 
five  departments  of  hearing,  sight,  and  so  on;  for  ex- 
ample, our  capacity  to  feel  pressure  gives  rise  to  a  dis- 
tinctive class  of  sensations,  and  so  the  pressure-sense  is 
spoken  of  as  a  sense  department;  in  like  manner,  we 
speak  of  sense  departments  corresponding  respectively  to 
the  capacities  for  feeling  warmths,  brightnesses,  sizes, 
sounds,  and  so  on,  it  being  evident  that  some  of  the  de- 

1  In    relation    to    the    question,    see    Russell's    Analysis    of   Mind    and 
Keyser's  review  of  It  in  The  Literary  Review  (N.  Y.  Evening  Post). 


THE    MATHEMATICS    OF    PSYCHOLOGY       371 

partments  are  sub-departments  of  others;  that  of  length, 
for  example,  is  a  division,  branch  or  species  of  the  generic 
department  of  size. 

Each  department  presents  three  problems — funda- 
mental problems.  It  is  easy  to  see  why  this  is  so,  and 
what  the  problems  are. 

Everyone  knows  that  some  lights  are  too  dim  to  be 
seen,  that  some  tastes  are  too  delicate  to  be  discerned, 
that  some  pressures  are  too  slight  to  be  felt,  that  some 
lengths  are  too  short  to  be  sensed,  and  so  on  for  the 
other  sense  departments.  The  form  of  my  statement  in- 
volves a  contradiction  in  terms  but  the  meaning  is  clear, 
and  that  is  enough.  Out  of  the  kind  of  facts  stated  arises 
one  of  the  experimenter's  problems.  The  problem  is : 
Given  a  sense  department,  to  determine  the  smallest 
amount  of  (the  appropriate)  stimulus  that  will  yield  a 
sensation. 

The  problem  just  stated  has  a  complement.  Every- 
one knows  that  a  stimulus  may  be  too  great,  as  well  as 
too  small,  to  produce  a  sensation.  Thus  in  the  sense  de- 
partment of  tones  there  is  no  sensation  answering  to 
100,000  vibrations  per  second;  a  pressure  may  be  so  great 
that  we  cannot  feel  it;  a  light  may  be  so  intense  as  to 
blind  us;  and  so  on.  Whenever  the  stimulus  exceeds  a 
certain  amount,  the  nerves  are  put  out  of  commission,  and 
the  scale  of  sensation  reaches  an  end.  Hence  the  prob- 
lem: Given  a  sense  department,  to  determine  the  maxi- 
mal stimulus  that  will  produce  a  sensation.  And  so  wc 
see  that  the  world  of  sensory  experience  possible  to  us 
humans  is  a  confined  world, — walled  in  by  the  surround- 
ing presence  of  limits,  a  lower  and  an  upper  limit  in  every 
department  of  sense.  The  two  problems  stated  arc  those 
of  determining  the  location  of  all  the  parts  of  the  wall. 


372  MATHEMATICAL    PHILOSOPHY 

The  third  one  of  the  experimenter's  fundamental 
problems  has  to  do  with  what  occurs  within  the  wall.  It 
is  this:  Given  a  sense  department  and  in  it  a  sensation 
corresponding  to  the  least  (or  greatest)  stimulus  that 
will  produce  it,  to  determine  how  much  the  stimulus  must 
be  increased  (or  decreased)  to  beget  a  new,  or  different, 
sensation. 

Some  Technical  Terms  and  Symbols. — In  the  litera- 
ture we  encounter,  as  you  probably  know,  the  equivalent 
terms — Threshold,  Limen,  Schwelle — introduced  by 
Herbart  about  a  hundred  years  ago,  and  the  symbols — ■ 
R,  L,  D,  T,  RL,  TL,  DL — whose  meanings  are  easy  to 
grasp.  R  comes  from  the  German  Rciz,  signifying  stimu- 
lus; L  stands  for  Limen,  or  threshold;  and  T  for  termi- 
nal; RL  denotes  initial  threshold, — the  least  stimulus  that 
will  yield  a  sensation;  TL  denotes  terminal  threshold, — 
the  greatest  stimulus  that  will  yield  a  sensation;  and  DL 
denotes  difference  threshold, — the  difference  between  the 
least  (or  greatest)  amounts  of  stimulus  that  correspond 
to  two  just  discernibly  different  sensations.  The  three 
fundamental  problems  may  accordingly  be  restated  thus: 
To  determine  in  each  sense  department  its  RL,  TL,  and 
DL. 

The  pioneers,  Weber  and  Fechner,  were  contempora- 
ries but  the  work  of  Weber  came  first.  He  dealt  mainly 
with  hearing  and  touch.  A  rough  statement  of  Weber's 
Law, — so  named  by  Fechner, — is  this:  The  increase  of 
stimulus  necessary  to  produce  a  change  of  sensation  is  not 
a  constant  difference,  but  is  a  constant  ratio  of  the  pre- 
ceding stimulus.  It  is,  you  notice,  concerned  with  the 
DL,  the  difference  threshold.  It  is  often  referred  to  as 
Fechner's  law  or  the  Weber-Fechner  law  or  the  psycho- 
physical  law.      It   is    to    Fechner, — whose   great   work, 


THE    MATHEMATICS    OF    PSYCHOLOGY      373 

Elemente  der  Psychophysik,  appeared  in  i860, — that  we 
owe  the  first  formulation  of  the  methods  which,  with 
many  modifications  and  improvements,  are  now  employed 
in  psychological  laboratories  throughout  the  world.  That 
is  why  Fechner  is  called  the  father  of  modern  psychology. 

Symbolic  Statement  of  the  Psychophysical  Law. — We 
are  not  here  concerned  to  deduce  the  law  nor  to  verify 
it,  but  to  state  it  in  symbols  and  to  examine  its  meaning. 

Consider  one  of  the  sense  departments,  say  that  of 
pressure.  Denote  by  Si  the  sensation  produced  by  a  cer- 
tain stimulus  Ri.  Next  suppose  the  stimulus  increased 
till  there  is  felt  a  sensation  S2  just  discernibly  different 
from  Sx.  Denote  the  new  stimulus  by  R2.  Suppose  the 
stimulus  to  be  again  increased  till  a  new  sensation  S3  is 
felt  that  is  just  distinguishable  from  S2,  and  denote  the 
third  stimulus  by  R3.  We  now  have  the  table  of  cor- 
respondents : 

Si *i 

02 R-2 

03 R3 

The  question  is:  how  are  the  Rys  related?  If  we  are  to 
suppose  them  related  at  all,  connected,  that  is,  by  some 
invariant  order,  or  law,  the  simplest  guess  would  be  that 

Rz  —  R2  —  R2  —  Ri- 

But  that  guess  would  be  wrong.  Experiment  shows  that 
it  is,  not  the  difference,  but  the  ratio  that  is  constant; 
that  is, 

R2     Rs 

Ri     R2 


374  MATHEMATICAL   PHILOSOPHY 

And  this  is  very  remarkable,  for  no  one  could  know  in 
advance  that  so  simple  a  relation  holds.  A  relation  so 
complex  as  not  to  admit  of  statement  in  a  finite  number 
of  words  may  exist  in  the  world  but,  for  scientific  pur- 
poses, such  a  relation  is  practically  equivalent  to  chaos,  to 
no  relation  at  all.  Denote  the  constant  ratio  by  K. 
Then  R2=KRi,  Rz=KR2.  As  K  exceeds  I,  we  may 
write  K  =  I  -\-r,  where  r  is  positive.     Then 

R2=R1(i+r) 
R3=R2(i+r)=R1(i+r)2. 

Denote  by  A  the  amount  of  stimulus  such  that  the 
amount  A{\  +r)  is  the  smallest  stimulus  that  will  yield  a 
sensation  in  the  sense  department  under  consideration; 
in  other  words,  A{\  -f-r)  is  the  RL  of  that  department.  Let 
the  sensation  corresponding  to  the  stimulus  A{\  +r)  be 
denoted  by  the  number  I.  The  absence  of  sensation  corre- 
sponding to  A  may  be  indicated  by  zero  (0).  The  experi- 
mental results  may  be  shown,  as  follows,  in  tabulated 
form. 

Sensations  Stimuli 

0 A 

i A(\+r) 

2 A{\+rf 


n A(i+r)f 


Of  course   the   final   continuation   marks   do   not   mean 
ad  infinitum,  for,  as  we  have  seen,  the  sensation  scale  has 


THE    MATHEMATICS    OF    PSYCHOLOGY      375 

an  upper  limit — an  end  as  well  as  a  beginning.     Observe 
that 

.           stimulus 
°gl+r ~A 

.           stimulus    . 
=  ~A '      gl+'e' 

Let  us  now  write  S  =  n,  R  =  stimulus -ri  and  C  =  log1+r  e. 
Then  we  have 

S  =  C  log  R 

Such,  then,  is  Fechner's  formulation  of  what  he  called 
the  Psychophysical  Law.  We  must  not  fail,  if  we  are  to 
understand  it,  to  note  very  carefully,  the  meaning  of  the 
symbols.  Observe  that  S,  the  initial  letter  of  the  word 
sensation,  denotes  nothing  but  a  cardinal  number.  Note 
that  the  constant  C  depends  on  r  and  that  r,  which  is 
the  same  for  the  sensations  of  a  given  sense  department, 
differs  for  different  departments.  Note  also  that  R  is 
not  the  stimulus  that  produces  S,  as  it  is  commonly  said 
to  be  (by  Professor  James,  for  example),  but  that  it  is 
the  quotient  of  that  stimulus  divided  by  A,  and  that  A 
depends  upon  the  sense  depaitment  under  investigation. 

The  sense  departments  that  have  been  most  investi- 
gated are  those  of  light,  muscular  sensation,  pressure, 
warmth  and  sound.  For  these  the  values  of  r,  as  reported 
by  Professor  Wundt,  are  as  follows: 

For  light,  r  =  about  tott 

For  muscular  sense,  r  =  about  tV 

For  pressure,  r=  about  ^ 

For  warmth,  r  =  about  § 

For  sound,  r  =  about  -| 


376  MATHEMATICAL   PHILOSOPHY 

In  other  departments,  where  investigation  is  more  dif- 
ficult, there  is  wide  divergence  in  the  results  that  have 
been  reported. 

The  Literature  of  the  Law. — "  Those,"  says  James, 
"  who  desire  this  dreadful  literature  can  find  it."  The 
best  of  it  is  cited  under  the  caption,  "  Weber's  Law," 
in  the  eleventh  edition  of  The  Encyclopedia  Britannica. 
For  an  excellent  introduction  to  the  methods  of  quantita- 
tive psychology,  I  have  pleasure  in  referring  to  Titchener's 
Experimental  Psychology. 

As  to  the  Validity  of  the  Law. — In  the  nature  of  things, 
no  law  can  be  shown  to  be  absolutely  valid  by  means  of 
experiment.  Respecting  Weber's  Law,  we  may  safely 
make  the  following  statements.  Experiment  has  shown 
it  to  be  approximately  valid  in  several  of  the  chief  depart- 
ments of  sense.  As  these  are  the  departments  most 
accessible  to  experiment,  it  may  be  that  the  law  will 
yet  be  found  to  be  approximately  valid  in  other  depart- 
ments. Such  validity  has  not  been  disproved  for  any 
department.  The  law  is  found  to  hold  best  in  the  mid- 
region  of  a  sense  scale;  that  is,  it  is  least  certain  near  the 
initial  and  terminal  thresholds.  This  fact,  however,  is 
consistent  with  the  assumption  that  the  lawisequally  valid 
throughout  the  scale,  for  it  is  plain  that,  near  the  begin- 
ning and  the  end  of  a  scale,  where  sensation  is  dim  because 
of  defect  or  excess  of  stimulus,  the  distinctions  of  dif- 
ferent sensations  are  more  difficult  to  detect  and  record. 

What  is  Measured. — Fechner  calls  the  law  of  psycho- 
physics  a  Maasformel.  But  what  is  it  that  is  measured? 
What  is  the  magnitude?  According  to  Fechner  it  is 
sensation.     He  says: 

Our  measure  of  sensation  amounts  to  this:  that  we 
divide  every  sensation  into  equal  parts,  that  is,  into 


THE    MATHEMATICS    OF    PSYCHOLOGY      377 

equal  increments  out  of  which  it  is  built  up  from  the 
zero  of  its  existence,  and  that  we  regard  the  number 
of  these  equal  parts  as  determined  by  the  number  of 
the  corresponding  variable  increments  of  stimulus  that 
are  able  to  arouse  the  equal  increments  of  sensation, 
just  as  if  the  increments  of  stimulus  were  the  inches 
upon  a  yard-stick. 

It  is  evident  that,  in  Fechner's  view,  two  just  discernibly 
different  sensations,  belonging  to  a  same  sense  depart- 
ment, differ  by  a  sensation-unit  (of  a  sort  characteristic 
of  such  department).  According  to  him,  a  sensation 
denoted  by  n  in  the  foregoing  table  is  the  sum  if  n  equal 
sensation-units,  that  is,  n  times  the  sensation  denoted  by 
the  number  I  in  the  table.  Thus,  according  to  Fechner's 
interpretation  of  the  experimental  facts,  sensation  in- 
creases as  the  terms  in  the  arithmetic  progression, 
I,  2,  3,  ...  ,  while  the  series  of  corresponding  stimuli  is 
an  increasing  geometric  progression.  And  this  logarith- 
mic correspondence  between  the  two  progressions  Fechner 
regarded  as  the  law  of  correlation  between  mind  and  matter, 
between  the  psychical  world  and  the  physical  world.  He 
thus  judged  that  he  had  made  a  very  great  discovery  and 
naturally  spoke  of  it  with  a  feeling  of  triumph.  Fechner 
was  at  once  physicist,  mathematician,  poet,  dreamer  and 
mystic — a  magnanimous  man  who  believed  that  all  the 
animals,  the  plants,  the  earth  and  the  stars  have  souls. 
For  the  considerations  that  led  him  to  this  noble  belief, 
see  his  works  entitled  Nanna  and  Zendavesta.  In  his  view, 
the  souls  of  human  beings  are,  in  the  scale  of  being,  inter- 
mediate to  the  souls  of  plants  and  the  souls  of  stars,  the 
latter  being  likened  by  him  to  angels.  God,  he  taught,  is 
the  soul  of  the  universe,  and  the  uniformities  that  we  call 
natural  laws  are  simply  the  ways  of  God.  This  view  of 
things  he  called  the  "  daylight  "  view  of  the  world   in 


378  MATHEMATICAL   PHILOSOPHY 

contrast  with  the  "  night  view  "  of  materialism.  It  is 
not  to  small  men,  but  to  great  ones,  that  this  sublime  con- 
ception of  our  universe  appeals.  Even  so  hard-headed 
a  scientific  man  as  Bernhard  Riemann,  whose  philosophic 
Fragments  I  have  already  alluded  to,  there  speaks  of  the 
conception  with  deep  interest  and  grave  respect.  And, 
as  you  may  be  interested  to  know,  Professor  William  James 
has  dealt  with  the  same  subject  in  one  of  his  latest  writings. 
I  refer  to  an  article,  entitled  "  The  Earth  Soul,"  which 
appeared  in  the  Hibbert  Journal,  January,  1909. 

Fechner's  view  that  a  given  sensation  is  a  sum  of 
sensation  units  has  not  ultimately  found  favor  with 
psychologists.  Why  not?  Mainly  because  a  sensation 
is  not  felt  or  sensed  as  a  sum  of  sensations  or  of  sensation 
units,  and  the  question  relates  to  sensations,  not  as  they 
may  appear  mediately  in  our  reflection  upon  them,  but  as 
they  appear  immediately  in  feeling.  A  sensation  of  bright- 
ness, for  example,  is  not  felt  to  be  composed  of  so-and-so 
many  units  of  brightness.  A  feeling  of  pink,  says  James, 
is  surely  not  a  portion  of  our  feeling  of  scarlet. 

And  so  the  question  recurs:  What  is  it  that  the  experi- 
ments have  measured?  Or  approximately  measured,  as 
we  ought  to  say,  for  it  is  evident  that  nothing  ever  is  or 
can  be  measured  with  absolute  precision  experimentally. 
Perhaps  we  may  say  that  what  is  measured  is  a  human 
sensorium's  discriminative  sensibility  to  stimulus.  Our 
sensibility  does  not  detect  every  difference  of  stimulus. 
It  does  detect  some  differences.  By  experiment  these  have 
been  approximately  ascertained  for  several  departments 
of  sense,  and  the  psychophysical  law  is  an  approximately 
accurate  statement  of  the  way  in  which  they  are  related. 

Sensation  as  a  Function  of  Stimulus.— In  the  mathe- 
matical meaning  of  the  term  "  function,"  sensation  is  a 


THE    MATHEMATICS    OF    PSYCHOLOGY      379 

function  of  stimulus  within  the  interval  between  the 
initial  and  the  terminal  thresholds  of  any  department. 
For  to  any  amount  of  stimulus  in  such  an  interval,  there 
corresponds  a  sensation.  That  is  to  say,  in  no  such 
interval  have  there  been  found  any  blind  spots  or  gaps 
or  regions  or  points  where  the  sensorium  fails  to  respond. 
The  question  arises :  Is  sensation,  or  the  intensity  of  sensa- 
tion, a  continuous  function  of  stimulus?  Most  psycholo- 
gists answer  yes.  Among  these  may  be  mentioned,  for 
example,  Titchener,  Ward  and  Stout.  James  has 
answered  no.  The  men  named  leave  one  in  doubt 
whether  they  know  precisely  what  is  meant  by  a  con- 
tinuous function.  Let  us  recall  to  mind  the  idea  of 
functional  continuity.  Let  us  remember  that  if  f(x)  is 
to  be  a  continuous  function  of  a  real  variable  x  in  an 
interval  having  a  for  its  beginning  and  b  for  its  end,  the 
following  conditions  must  be  satisfied : 

(i)  If  x'  be  any  value  of  x  in  the  interval,  then/(V)  = 
some  definite  value. 

(2)  Limit  f(xf±Ax)  =/(#')  as  Ax  approaches  zero. 
Condition  (1)  is  indeed  included  in  (2)  but  it  is  helpful  to 
state  it  explicitly.  Now  we  know  that  the  ordinary  func- 
tion, y=c\ogx,  is  continuous  throughout  any  interval 
not  containing  the  value,  x=0.  But  is  the  Fechner  func- 
tion, S  =  C  log  R,  a  continuous  function  of  R?  No;  for 
consider  a  stimulus  greater  than  A{i+r)n  and  less  than 
A{\  -\-r)n  ;  compare  the  corresponding  sensation  S  with 
that  denoted  by  n  (in  the  above-given  table)  and  then 
compare  it  with  that  denoted  by  n  + 1 ;  in  the  first  compari- 
son S  appears  to  be  n;  in  the  second,  it  appears  to  be 
n-\- 1.  Condition  (1)  is,  you  see,  not  satisfied.  Then,  of 
course,  condition  (2)  is  not  satisfied.  Moreover,  when  we 
speak  of  fix)  as  a  continuous  function,  we  imply  that  the 


380  MATHEMATICAL    PHILOSOPHY 

variable  x  can  vary  continuously.  In  the  Fechner  func- 
tion, however,  the  variable  R,  which  means  stimulus 
divided  by  A,  can  not  thus  vary.  For  weights,  pressures, 
the  number  of  air  or  of  ether  vibrations  per  second,  etc., 
when  they  vary,  vary  discontinuously;  the  changes  may 
indeed  be  small,  but  they  are  finite,  and  they  occur  as  a 
leap  or  bound.  It  thus  appears  that  to  debate  whether  or 
not  sensation  is  a  continuous  function  of  stimulus  is  to 
engage  in  the  rather  meaningless  exercise  of  discussing 
whether  a  function  of  a  discontinuous  variable  is  a  con- 
tinuous function  of  it. 

The  Fallacy  of  the  Tangent  Galvanometer  Experiment. — 
Imagine  a  coil  or  ring  of  copper  wire  placed  in  the  plane 
of  the  magnetic  meridian.  At  its  center  is  suspended  a 
small  magnetic  needle  so  it  may  turn  in  a  horizontal 
plane.  In  its  initial  position  it  points  towards  the  mag- 
netic pole.  A  current  of  electricity  passed  through  the 
wire  will  cause  the  needle  to  turn  by  an  amount  depending 
on  the  strength  of  the  current.  Now  notice  how  hand- 
somely the  behavior  of  the  needle  resembles  that  of  sensa- 
tion. The  electric  current  plays  the  role  of  stimulus. 
Owing  to  the  presence  of  friction  in  the  turning  of  the 
needle,  a  certain  small  strength  of  current  is  necessary  to 
start  the  needle.  This  amount  may  be  likened  to  the 
"  initial  threshold."  And  there  is  a  kind  of  "  terminal 
threshold,"  too,  for  no  finite  strength  of  current  can  bring 
the  needle  to  an  angle  of  900  with  the  meridian  plane. 
When  the  needle  starts,  it  leaps  to  a  certain  position  and 
there  remains,  if  the  current  be  steady,  till  the  current 
has  been  increased  by  a  certain  amount,  when  the  needle 
leaps  again,  and  again  remains  in  its  new  position  if  the 
current  be  steady,  and  so  on  and  on.  And  so  we  see  there 
is  here  involved  a  kind  of  "  difference  threshold."     The 


THE    MATHEMATICS    OF    PSYCHOLOGY      881 

like  phenomena  are,  of  course,  observed  in  the  behavior 
of  a  pair  of  scales  for  weighing.  It  is  evident  that,  owing 
to  friction,  the  tangent  of  the  needle's  angle  with  the 
meridian  plane,  though  it  is  a  function  of  the  current's 
strength,  is  not  a  continuous  function  of  it.  It  is  common, 
however,  to  idealize  the  situation,  by  disregarding  the 
frictional  effect,  and  to  say  that  the  function  is  continuous, 
though  as  a  matter  of  fact  it  is  not.  Misled  by  such 
considerations  certain  psychologists  have  argued  falla- 
ciously as  follows:  The  tangent  of  the  needle's  angle  really 
is,  they  say,  a  continuous  function  of  the  current's 
strength.  They  admit  that  it  does  not  appear  to  be  so, 
but  that,  they  say,  is  because  the  fact  or  the  law  is 
"  masked  "  by  the  presence  of  friction.  And  then  they 
contend  that  the  situation  is  essentially  the  same  in  the 
case  of  sensation.  Sensation,  they  say,  is  indeed  a  con- 
tinuous function  of  stimulus,  though  it  does  not  appear 
to  be  such.  For,  they  say,  the  continuity  is  "  masked  " 
by  what  they  call  the  "  frictional  effect  "  or  the  opposition 
of  some  chemical  or  mechanical  or  physical  resistance 
offered  by  the  sensorium  to  the  action  of  stimulus.  The 
fallacy  is  obvious.  It  consists  in  ignoring  some  of  the 
facts.  The  question  is  not  whether  the  functions  under 
consideration  would  be  continuous  if  there  were  no 
friction  or  "  frictional  effects,"  but  whether  they  are  in 
fact  continuous  in  a  world  where  friction  and  "  frictional 
effects  "  persist  as  part  of  reality. 

The  Number  of  Possible  Sensations  Finite. — How  many 
different  sense  departments  are  there?  I  know  of  no  way 
to  prove  that  the  number  is  finite,  but  the  assumption 
that  the  number  is  finite  seems  to  be  very  probably  correct. 
Let  us  make  the  assumption.  We  know  that  the  number 
of  discernibly  different  sensations  that   any  department 


382  MATHEMATICAL    PHILOSOPHY 

admits  is  finite.  It  follows  that  the  total  number  of 
different  sensations  of  which  a  human  being  is  capable 
is  a  finite  number.  I  am  not  aware  that  this  important 
consideration  has  been  recognized  in  the  literature  of 
psychology.  It  would  not  be  strange  if  it  has  not,  for 
the  distinction  of  finite  and  infinite,  though  it  is  very  im- 
portant scientifically  and  in  some  of  its  connections  is 
awe-inspiring,  has  not  yet  gained  much  intelligent  recogni- 
tion outside  the  circle  of  mathematicians.  What  I  wish 
to  signalize  here  is  this:  the  fact  that  the  ensemble  of 
possible  concepts  is  infinite  and  the  fact  that  the  ensemble 
of  possible  sensations  or  percepts  is  finite  together  confront 
with  a  difficult  and  important  problem  those  psychologists 
who  hold  that  our  mental  life  is  based  on  sensation  in 
the  sense  that  all  ideas  arise  out  of  sensation.  For 
others  the  problem  does  not  exist.  Plato,  for  ex- 
ample, held  ideas  to  be  eternal,  existing  before,  during 
and  after  sensation  or  perception,  and  that  the  world 
of  sensation  or  perception  is  only  an  imperfect  and 
transitory  imitation  of  the  eternal  world  of  ideas  or 
concepts. 

Sense  Continua. — We  have  seen  that  the  results  of 
experiment  do  not  warrant  us  in  saying  that  sensation  is 
a  continuous  function  of  stimulus  in  the  mathematical 
sense  of  the  term.  Nevertheless,  it  will  be  convenient  to 
speak  of  physical  or  experiential  or  sensible  or  sense 
"  continua  "  and  we  shall  do  so  but  we  shall  thereby 
mean  merely  that  to  any  amount  of  stimulus  between  the 
initial  and  terminal  thresholds  of  a  sense  department 
there  corresponds  a  sensation  in  the  department.  We 
must  at  no  time  confound  this  meaning  of  continuous  with 
the  mathematical  meaning  of  the  term.  With  this  under- 
standing we  will   speak   of  the   "  continuum  "   of  pres- 


THE    MATHEMATICS    OF    PSYCHOLOGY      383 

sures  or  of  weights  or  of  sizes  or  of  sounds  and  so  on  for 
the  other  departments  of  sense. 

A  Remarkable  Property  of  Sense  "  Continual — Owing 
to  the  presence  of  difference  thresholds  in  sense  depart- 
ments, the  sense  continuum  of  any  department  possesses 
a  remarkable  property.  It  is  this:  If  Si,  S2,  S3  are  sensa- 
tions corresponding  to  three  different  amounts,  Rly  R2y  Rs, 
of  stimulus,  it  may  happen  that  no  two  of  the  S's  are  dis- 
tinguishable from  each  other  and  it  may  happen  that  Si 
is  indistinguishable  from  So,  and  that  S2  is  indistinguish- 
able from  S3  but  that  nevertheless  Si  and  S3  are  dis- 
tinguishable from  each  other.  So  that  in  a  given  depart- 
ment we  may  have  three  S's  (three  in  the  sense  of  their 
being  produced  by  three  different  amounts  of  stimulus) 
such  that 

(i)    Si=S2,      02  =  03,      Si=S3; 

and  we  may  have  three  such  that 

(2)  Si  =  S2,     So  =  S3,     Si^Ss; 

of  course  there  are  other  possibilities.  Thus  if  the  sense 
"  continuum  "  be  that  of  pressures,  then,  if  the  S's 
correspond  respectively  to  10,  iof  and  1 1  grams,  they 
satisfy  relations  (1);  but  if  they  correspond  to  10,  n  and 
12  grams,  they  satisfy  relations  (2).  The  matter  may, 
of  course,  be  exemplified  in  other  sense  "  continua." 
Thus  in  the  sense  "  continuum  "  of  lengths  the  sensations 
produced  by  (say)  three  pencils  of  three  different  lengths 
may  satisfy  relations  (1)  or  (2). 

Another  Aspect  of  the  Matter. — This  matter  has  another 
striking  and  important  aspect.  If  sensation  be  allowed  to 
judge,  then  we  should  say  that  a  stimulus  Q  and  a  stimulus 
Q'  (of  the  same  kind  as  Q)  arc  equal  if  the  corresponding 


384  MATHEMATICAL    PHILOSOPHY 

sensations,  S  and  S',  are  indistinguishable,  and  that  the 
0's  are  unequal  when  the  S's  are  distinguishable.  Accord- 
ingly, if  sensation  be  the  judge,  then  three  quantities, 
0i>  02,  03,  belonging  to  a  given  sense  "  continuum,"  may 
be  such  as  to  satisfy  the  relations 

(3)  01=02,        02=03,        01=03, 

or  such  as  to  satisfy  the  relations 

(4)  01=02,        02=03,        01^03. 

Now,  as  we  noted  in  the  preceding  lectures,  the  series 
(2)  of  relations  or  the  series  (4)  violates  the  logical  law  of 
Contradiction  (or  of  Non-contradiction,  as  it  is  sometimes 
called).  What  of  it?  In  response  we  have  only  to  reflect 
upon  the  role  of  that  law  in  the  realm  of  rational  life. 
The  law  is  indispensable  to  logical  thinking,  to  science, 
to  the  very  life  of  intellect.  No  doubt  a  pig  or  other 
animal  may  treat  some  object  0\  as  if  it  were  the  same 
as  some  different  object  O2,  then  react  to  O2  as  if  it  were 
the  same  as  some  third  object  O3,  and  then  react  to  0\ 
as  if  it  were  different  from  O3;  and  this  the  animal  may  do 
without  feeling  any  sense  of  shock  or  surprise.  This 
sense  of  shock  is  a  human  experience, — an  idiosyncrasy 
of  a  rational  being, — a  mark  of  man.  Might  we  not  em- 
ploy it  as  the  definition  of  man?  Instead  of  saying  with 
Plato  that  man  is  a  featherless  biped,  would  it  not  be 
better  to  define  man  as  the  creature  that  is  capable  of 
feeling  the  shock  of  contradiction  (2)  or  (4)  ?  Such  a 
definition  would  have  the  merit  of  excluding  from  the 
genus  homo  some  featherless  bipeds  even  if  it  did  not 
include  any  quadrupeds.  However  this  may  be,  we  are 
literally  driven,  on  pain  of  intellectual  or  logical  extinction, 
to  say,  in  the  case  of  the  relations  (4),  that  Qi  9^Q-i  or  that 


THE    MATHEMATICS    OF    PSYCHOLOGY      385 

Q'iT^Qz  or  that  both  of  these  inequalities  subsist  despite 
the  verdict  of  sensation  to  the  contrary.  But  what  does 
that  mean?  It  means  that  we  are  driven  to  assume  that 
there  are  quantities  of  magnitude  so  small  as  to  be  insen- 
sible, too  small  to  be  sensed  or  felt  or  perceived.  And 
what  does  that  mean?  It  means— and  the  answer  is 
fundamentally  important— that  we  are  driven  to  posit  or 
postulate  the  existence  of  purely  conceptual  quantities  or 
amounts  of  magnitude.     And  we  do  it. 

Properties  of  the  Conceptual  Magnitudes. — We  must 
not  fail  to  note  some  of  the  properties  with  which  our 
human  minds  have  been  constrained  to  endow  such  con- 
ceptual magnitudes.  If  we  were  to  suppose  our  discrimi- 
native sensibility  to  be  by  some  means  so  increased  or 
refined  as  to  bring  the  assumed  insensible  quantity 
Qi^Qz  (tne  difference  between  the  sensible  quantities 
Qi  and  Qo)  well  within  the  domain  of  the  increased  sensi- 
bility, we  have  no  reason  to  doubt  that  the  old  phenome- 
non would  recur  therein;  we  should,  that  is,  expect  to 
find  quantities  q\y  q>ly  qz  within  the  interval  Q\<*>Q2  such 
that 

qi=q2,     q2=qs,     qi^qs", 

that  is  to  say,  fatal  violence  to  the  mentioned  law  of 
logic  would  remain.  And  so  it  would  did  we  suppose 
our  sensibility  to  be  again  increased  so  as  to  render 
sensible  the  newly  postulated  quantity,  qx^qy,  and  so 
on  and  on.  Observe  attentively  that  the  indicated  process 
of  intercollating  ever  smaller  and  smaller  insensible- 
quantities  is  precisely  like  that  by  which  symbols  called 
rational  fractions  are  interlarded  between  the  integers 
and  then  other  rational  fractions  are  inserted  between  the 
former  ones,  and  so  on  and  on.     Observe  also  that,  in 


386  MATHEMATICAL    PHILOSOPHY 

order  by  this  means  to  rescue  the  law  of  non-contradiction 
from  the  violence  of  the  relations  (4),  it  would  be  necessary 
to  suppose  our  discriminative  sensibility  to  be  increased 
till  the  difference  between  quantities  (of  magnitude) 
corresponding  to  any  two  rational  fractions,  however  slight 
their  difference,  would  be  sensibly  discernible,  that  is, 
sensation  would  have  to  be  a  continuous  function  of 
stimulus  where  continuity  signified  continuity  defined  in 
the  domain  of  rational  numbers.  The  definition  would 
be  the  same  as  that  above  recalled  except  that  the  function 
and  its  argument  would  be  restricted  to  rational  values. 

No  such  endless  refining  of  sensibility  has  occurred 
nor  is  it  possible,  but  we  know  that  in  the  long  course  of 
time  the  various  kinds  of  sensible  magnitude, — the  magni- 
tudes revealed  by  or  in  sensation, — have  gradually  been 
submitted  to  a  conceptualizing  process  that  is,  in  a  very 
important  respect,  virtually  equivalent  to  the  refining 
process  in  question.  The  various  kinds  of  sensible  magni- 
tude,— the  various  sense  "  continua  "  presented  respect- 
ively in  various  sense  departments  of  sound,  color,  weight, 
taste,  warmth,  duration,  hardness,  spatial  extent,  velocity, 
acceleration,  etc., — have  all  of  them,  in  the  course  of  the 
centuries,  got  replaced  in  our  thinking  with  corresponding 
insensible  or  conceptual  magnitudes  having  the  structure 
of  the  system  of  rational  numbers.  The  process  has  been 
partly  conscious  and  partly  unconscious.  This  system, 
you  know,  is  what  has  been  sometimes  called  the  mathe- 
matical continuum  of  first  order,  as  by  the  late  Professor 
Henri  Poincare,  for  example,  in  Science  and  Hypothesis. 
So  we  may  say  that  one  of  the  great  achievements  of  the 
human  intellect  in  its  dealing  with  the  magnitudes 
revealed  in  sensation  has  been  the  substitution,  for  such 
sense  "  continua,"  of  conceptual  magnitudes  or  conceptual 


THE    MATHEMATICS    OF    PSYCHOLOGY      387 

"  continua  "  of  the  type  of  the  mathematical  "  con- 
tinuum "  of  the  first  order.  As  already  pointed  out,  the 
great  replacement  or  substitution  has  been  logically 
coerced  by  the  mentioned  contradiction  that  inheres  in 
the  "  continua  "  of  sense.  The  aim,  which  has  been  con- 
scious or  unconscious,  and  the  issue  have  been  emanci- 
pation, intellectual  harmony,  increase  of  freedom  from 
logical  discord:  we  know  that,  if  qi,  qo,  qz  be  quantities  of 
a  conceptual  magnitude  or  "  continuum  "  of  the  type  in 
question,  and  if  qi=q2,  and  ^2=^3,  then  q\=qz  without 
exception. 

The  Need  of  Further  Emancipation. — But  the  replace- 
ment mentioned  is  not  enough.  For  starting  with  the 
kind  of  conceptual  magnitudes  in  question,  we  are  destined 
to  encounter  contradictions  or  discords  of  another  sort. 
This  fact  may  be  shown  as  follows:  Consider  a  sensible 
line.  It  is  for  sensibility  a  thing  like  a  rope  or  a  cable 
or  a  chalk  mark.  In  it  inheres  the  old  contradiction  (4). 
Now  suppose  it  replaced  by  a  corresponding  conceptual 
magnitude  of  the  type  of  the  first  order  "  continuum." 
The  new  thing  can  be  made  as  thin  and  as  narrow  as  the 
difference  between  any  two  rational  numbers.  But  this 
difference  can  be  made  to  approach  as  near  as  we  please 
to  zero.  Taking  the  limit,  we  have  a  line  having  only 
length,  the  thickness  and  the  width  being  zero.  Such 
a  line  is,  we  say,  geometric;  it  is  not  sensible,  but  is  purely 
conceptual.  Let  us  now  take  two  such  sensible  lines 
having  a  (sensible)  bulk  or  piece  in  common,  as  indicated 
in  Fig.  31,  and  let  us  suppose  them  submitted  to  the  con- 
ceptualizing process  above  indicated.  The  result,  we  say, 
is  two  geometric  lines  and  a  geometric  point  common  to 
them,  the  point  being,  we  say,  the  limit  of  the  bulk  that 
the  lines  have  in  common  as  sensible  lines  and  also  as 


388  MATHEMATICAL   PHILOSOPHY 

conceptual  lines  before  passing  to  the  limit.  We  are 
thus  led  to  believe  that  any  two  conceptual  lines  that 
cross  one  another  have  a  point  in  common.  (In  this 
connection  see  the  book  of  Poincare  above  referred  to.) 
Such,  however,  is  not  always  the  case  if  the  lines  are 
"  continua  "  of  only  first  order,  that  is,  if  they  have 
points  that  correspond  only  to  rational  numbers  or,  as 
we  say,  to  rational  coordinates.  It  is  sufficient  to  con- 
sider the  crossing  of  a  circle  and  a  straight  line  through 
its  center,  if  either  or  both  of  them  be  supposed  to  have 
no  points  except  such  as  have  rational  coordinates.     For 


Fig.  31. 

let  at2+v2  =  i  and  y=x  be  respectively  such  a  circle  and 
line.  If  we  undertake  to  solve  for  their  common  points, 
we  immediately  find  that  they  have  none;  for  we  obtain 
x  =  i  :  V2  and  y  =  i  :  V2;  but  these  numbers  are  not 
rational  and  hence  do  not  represent  a  point  common 
to  the  crossing  line  and  circle.  It  might  be  supposed  that 
such  a  circle  could  have  only  a  finite  number  of  points 
and  that  their  ensemble,  if  plotted,  would  not  look  like  a 
circle.  It  is  easy,  however,  to  prove  that  the  number 
of  the  points  is  infinite  and  that  they  constitute  a  dense 
set,  a  set,  that  is,  such  that  between  any  two  of  them 
there  is  another  one.  Obviously,  one  of  the  points  is  the 
point  (0,  1).  Suppose  x^O.  Then  since  x  and  y  are  to 
be  rational,  y  must  be  of  the  form,  y  =  1  —  rx,  where  r  is 


THE    MATHEMATICS    OF    PSYCHOLOGY      380 

rational.     Substitute    in   x2-\-y2  =  \.     We   thus   get   x2  + 
(i  —  rx)2  =  i,  whence 

2r 


x  = 


anc 


i+r2 

i^r2 
I  +r2 

Thus  it  is  seen  that  our  circle,  which  by  hypothesis  con- 
tains no  points  except  such  as  have  rational  coordinates, 
yet  contains  an  infinite  number  of  points — one  for  each 
rational  value  of  r.  It  is  evident  also  that,  given  any  one 
of  them,  there  is  another  one  as  near  to  it  as  we  please. 


— X1 


Fig.  32. 

Another  illustration:  if,  operating  in  the  conceptual 
"  continuum  "  of  rational  lengths,  we  affirm  the  existence 
of  a  square,  we  find  the  "  continuum  "  contains  no 
quantity  or  magnitude  to  connect  the  diagonally  opposite 
corners.  For,  if  the  length  of  the  square's  side  be  J, 
the  length  of  the  diagonal  dy  if  there  were  a  diagonal, 
would  be  d=sV2;  but  as  s  is  rational,  sVz  is  not  rational 
and  so  it  is  not  a  length  in  the  "  continuum  "  of  operation. 


390  MATHEMATICAL   PHILOSOPHY 

The  discovery  of  the  incommensurability  of  the  side  and 
the  diagonal  of  a  square,  which  was  a  very  great  discovery, 
is  commonly  attributed  to  Pythagoras.  It  is  said  that  the 
discovery  made  him  unhappy.  Why?  Because  he  was 
a  union  of  scientific  man  and  religious  mystic,  who 
taught  that  Number  is  the  essense  of  all  things.  But  here 
was  something, — the  ratio  of  the  side  and  diagonal  of  a 
square, — that  no  number  recognized  by  him  and  his  sect 
could  express.  This  fact,  should  his  religious  followers 
find  it  out,  might  disturb  or  destroy  their  faith.  Accord- 
ingly, so  it  is  said,  Pythagoras  decided  to  keep  the  fact  a 
secret  among  a  few  of  his  leading  disciples,  and  "  one  of 
their  number,  Hippasos  of  Metapontion,  is  even  said  to 
have  been  shipwrecked  at  sea  for  impiously  disclosing  the 
terrible  discovery  to  their  enemies."  You  should  not 
fail  to  read  what  Bertrand  Russell  has  said  on  this  subject 
in  his  recently  published  book  entitled  Our  Knowledge  of 
the  External  World  as  a  Field  for  Scientific  Method  in 
Philosophy.  Of  philosophic  method  in  science  we  hear 
much — the  phrase  is  familiar  if  the  thing  itself  is  not. 
But  what,  pray,  is  meant  by  "  scientific  method  in 
philosophy"?  That  is  what  Mr.  Russell  aims  to  tell 
us,  in  outline,  in  this  book.  It  is  in  the  main  a  rough, 
preliminary,  semi-popular  sketch  or  adumbration  of  a 
method  that  is,  I  am  told,  being  employed  by  Mr.  White- 
head in  minute  detail  in  his  preparation  of  the  fourth 
volume  of  the  Principia  Mathematica.  I  commend  the 
book  to  your  serious  attention.  Whoever  it  was  that  dis- 
covered incommensurables,  it  is  certain  that  their  exist- 
ence was  known  to  the  Greeks,  as  we  learn,  for  example, 
in  the  Metaphysics  of  Aristotle.  It  is  a  pleasure  to  be  able 
to  say  that  Aristotle  has  at  length  learned  to  speak 
English.     I    refer   to   W.    D.    Ross's   translation   of  the 


THE    MATHEMATICS    OF    PSYCHOLOGY      391 

Metaphysics  and  to  the  English  translations  of  Aristotle's 
other  works  published  or  being  published  by  The  Claren- 
don Press.  On  page  983a  of  Ross's  translation  we  find 
the  following: 

For  all  men  begin,  as  we  said,  by  wondering  that 
the  matter  is  so  (as  those  who  have  not  yet  perceived 
the  explanation  marvel  at  automatic  marionettes) — 
whether  the  object  of  their  wonder  be  the  solstices  or 
the  incommensurability  of  the  diagonal  of  a  square 
with  the  side;  for  it  seems  wonderful  to  all  men  that 
there  is  a  thing  which  cannot  be  measured  even  by  the 
smallest  unit.  But  we  must  end  in  the  contrary  and, 
according  to  the  proverb,  the  better  state,  as  is  the 
case  in  these  instances  when  men  learn  the  cause; 
for  there  is  nothing  which  would  surprise  a  geometrician 
so  much  as  if  the  diagonal  turned  out  to  be  com- 
mensurable. 

That  rendering  is  hard  to  beat.  It  is  true  that  to 
make  the  thought  of  the  Stagirite  quite  agree  with  the 
modern  mathematical  conception  of  the  matter  it  would 
be  necessary  to  replace  the  phrase,  "  by  the  smallest  unit  " 
by  some  such  expression  as,  by  any  unit  however  small, 
for  there  is  no  such  thing  as  the  smallest  unit,  just  as, 
zero  being  excluded,  there  is  no  smallest  rational  number 
(nor  indeed  a  smallest  irrational  number).  The  transla- 
tion, however,  is  excellent.  Do  but  compare  it  with  the 
following  translation  taken  from  the  Metaphysics  of 
Bohn's  Classical  Library. 

For,  indeed — as  we  have  remarked — all  men  com- 
mence their  inquiries  from  wonder  whether  a  thing  be 
so,  as  in  the  case  of  the  spontaneous  movements  of 
jugglers'  figures,  to  those  who  have  not  as  yet  specu- 
lated into  their  cause;  or  respecting  the  solstices,  or  the 
incommensurability  of  the  diameter;   for  it  seems  to  be 


392  MATHEMATICAL    PHILOSOPHY 

a  thing  astonishing  to  all,  if  any  quantity  of  those 
that  are  the  smallest  is  not  capable  of  being  measured. 
But  it  is  necessary  to  draw  our  inquiry  to  a  close  in  a 
direction  the  contrary  to  this,  and  towards  what  is 
better,  according  to  the  proverb.  As  also  happens  in 
the  case  of  these,  when  they  succeed  in  learning  those 
points;  for  nothing  would  a  geometrician  so  wonder  at, 
as  if  the  diameter  of  a  square  should  be  commensurable 
with  its  side. 

No  one  who  had  grasped  the  author's  thought  even  fairly 
well  could  have  written  that.  And  these  two  specimens 
are  typically  representative:  they  serve  to  exemplify  the 
comparative  merits  of  the  two  translations  as  wholes. 

The  Grand  Continuum. — On  first  encountering  the  sort 
of  contradiction  or  discord  dealt  with  a  moment  ago,  our 
minds  are  surprised  and  shocked  because  we  cannot  but 
believe  that  there  must  be,  or  ought  to  be,  a  kind  of 
magnitude  such  that  a  definite  part  or  amount  of  it  just 
reaches  from  a  corner  to  the  diagonally  opposite  corner 
of  any  given  square  and  because  we  cannot  but  feel  that 
geometric  lines  or  curves  ought  to  be  so  conceived  that, 
if  they  cross,  they  intersect,  or  have  a  point  (or  points) 
in  common.  What  has  the  human  mind  done  about  it? 
What  has  it  done  to  secure  release  from  the  kind  of  dis- 
cord in  question  and  so  to  enlarge  the  sphere  of  intellectual 
harmony?  What  it  has  done  is  this:  it  has  assumed  or 
created  the  kind  of  magnitude  required.  This  new  sort 
of  magnitude  is,  of  course,  not  sensible;  it  is  conceptual, 
but  it  is  not  of  the  type  of  the  mathematical  continuum 
of  first  order,  for  it  is  in  this  type  that  the  difficulties 
to  be  overcome  have  their  roots;  the  structure  of  the  new 
variety  of  magnitude  is  patterned  on  the  structure  of  the 
mathematical  continuum  of  second  order:  this  latter 
continuum  is  the  mathematical  continuum  proper — the 


THE    MATHEMATICS    OF    PSYCHOLOGY      393 

"  Grand  Continuum  "  as  it  was  called  by  Professor 
Sylvester.  It  contains  such  symbols  as  V2,  <?,  -k  and 
countless  hosts  of  others  sandwiched  between  the  symbols 
constituting  the  first-order  continuum,  much  as  the 
rational  fractions  of  this  are  sandwiched  between  the 
cardinal  numbers. 

The  Grand  Continuum  has  been  a  subject  of  profound 
investigation  by  mathematicians,  especially  during  the 
last  half-century,  and  has  long  been  everywhere  an 
important  theme  of  university  instruction  in  what  is  called 
the  theory  of  the  real  variable.  This  instruction,  which 
has  found  its  way  into  numerous  text-books  on  function 
theory,  is  mainly  based,  directly  or  indirectly,  upon  three 
classical  expositions  of  the  matter.  I  refer  to  Dedekind's 
exposition,  which  has  been  translated  by  Beman  and 
Smith  and  with  another  of  the  author's  works  has  been 
published  under  the  title  Essays  on  Number;  to  that  by 
Georg  Cantor  in  his  creative  memoirs  on  Mannigfaltig- 
keitslehre  (or  M en  genie hre);  and  to  the  exposition  found 
in  the  works  of  Weierstrass.  For  our  present  purpose  it 
will  be  sufficient  to  remind  ourselves  briefly  of  one  way 
in  which  the  concept  of  the  Grand  Continuum  may  be 
formed  and  of  its  two  essential  or  definitive  marks.  Con- 
sider the  following  two  sequences  of  rational  numbers: 
first,  the  sequence  of  all  rational  numbers  such  that  each 
of  them  is  less  than  2;  second,  the  sequence  of  all  rational 
numbers  such  that  the  square  of  each  of  them  is  less  than  2. 
Each  of  the  sequences  approaches,  as  we  say,  a  definite 
somewhat  as  a  limit.  The  limit  of  the  first  is  2,  which  is 
rational;  the  limit  of  the  second  is  not  rational;  we  call 
it  irrational^  denote  it  by  the  symbol  V2,  and  say  that  this 
irrational  is  given  or  defined  by  the  sequence  (or  any 
other  sequence)  having  it  for  limit.     There  are  infinitely 


394  MATHEMATICAL    PHILOSOPHY 

many  such  sequences, — sequences  (of  rationals),  that  is — 
that  approach  perfectly  definite  somewhats  as  limits 
which  (limits),  however,  are  not  rational  numbers.  The 
total  ensemble  of  definites  thus  defined  or  definable  is  the 
system  of  irrational  numbers.  And  these,  taken  together 
with  the  rational  numbers  from  which  they  are  thus 
derived  or  derivable,  are  commonly  said  to  constitute  the 
system  of  real  numbers.  This  use  of  the  terms  "  rational," 
"  irrational  "  and  "  real,"  though  dictionally  somewhat 
unfortunate,  is  historically  justified.  I  need  not  say  that 
as  employed  in  mathematics,  these  terms  have  completely 
lost  whatever  metaphysical  connotation  they  may  once 
have  had:  rational  does  not  signify  reasonable;  nor 
irrational,  unreasonable;  nor  is  a  "  real  "  number  any 
more  real  metaphysically  than  is  any  other  sort  of  number. 
Well, — to  return  from  this  cautionary  digression, — it  is 
the  real  numbers  that  constitute  the  Grand  Continuum. 
Perhaps  it  were  better  to  say  that  the  system  of  real 
numbers  is  the  basal  instance  of  the  Grand  Continuum 
for  other  continua  essentially  like  it  are  derived  from  it 
as  the  model.  You  are  aware  that  it  is  common  to  give 
the  name  continuum  to  any  segment  of  the  Grand  Con- 
tinuum, where,  by  segment,  I  mean  any  two  real  numbers 
together  with  all  the  numbers  that  lie  between  them  in 
value.  Thus  the  numbers  zero  and  I,  with  the  real  num- 
bers between  them,  constitute  a  continuum  of  second 
order.  The  most  convenient  and  vivid  example  or 
representation  of  such  a  continuum  is  the  ensemble  of 
points  constituting  a  straight  line-segment  as  ordinarily 
conceived — as  conceived,  that  is,  in  such  a  way  that  by 
taking  an  arbitrary  point  of  the  line  for  origin  and  employ- 
ing an  arbitrary  unit  of  length  or  distance,  a  one-to-one 
correspondence  can  be  set  up  between  the  points  of  the 


THE    MATHEMATICS    OF    PSYCHOLOGY       395 

line  and  the  real  numbers.  Such  a  continuum  of  points 
or  of  real  numbers  is  a  linear  or  o/^-dimensional  contin- 
uum of  second  order.  The  ensemble  of  pairs  of  real 
numbers  or  the  ensemble  of  points  of  a  plane  (as  ordi- 
narily conceived,  in  analytic  geometry,  for  example)  is 
a  ^o-dimensional  continuum  of  second  order;  for  a  three- 
dimensional  one,  it  suffices  to  refer  to  the  ensemble  of 
triplets  or  triads  of  real  numbers  or  to  the  ensemble  of  the 
points  of  our  familiar  geometric  space  as  ordinarily  con- 
ceived. And  it  is  evident  that  second-order  mathe- 
matical continua  may  have  any  given  dimensionality 
whatever.  For  a  logically  much  more  refined  account  of 
the  system  of  real  numbers,  you  should  examine  Russell's 
Principles  and  especially  the  Principia.  I  am  giving  here 
but  a  sketch  of  the  usual  account. 

The  Definitive  Marks  of  a  Grand  Continuum. — What 
are  the  characteristic  or  definitive  marks  or  properties  of 
a  mathematical  continuum  of  second  order?  The  answer 
is:  an  ensemble  of  numbers  or  points  or  other  elements  is 
such  a  continuum  when  and  only  when  the  ensemble  is 
compendent  and  perfect.  These  are  technical  terms.  What 
do  they  mean?  Let  us  answer  in  terms  of  points.  An 
ensemble  of  points  is  compendent  (or  zusammenhangend 
as  the  Germans  say  or  connected  as  it  is  common  to  say 
in  English)  if  it  be  such  that,  given  any  two  points  of  it, 
it  is  possible,  by  stepping  only  on  points  of  the  ensemble, 
to  pass  from  one  of  the  given  points  to  the  other  by  a 
finite  number  of  steps,  where  each  step  is  equal  to  or  less 
than  some  previously  assigned  distance,  however  small. 
An  ensemble  of  points  is  perfect,  provided  it  be  identical 
with  the  ensemble  of  its  limit-  points,  where,  by  a  limit- 
point  of  an  ensemble,  is  meant  a  point  such  that  there  are 
points  of  the  ensemble  distant  from  the  given  point  by  an 


7 


396  MATHEMATICAL   PHILOSOPHY 

amount  less  than  any  prescribed  distance,  however  small. 
It  is  easy  to  see  that  the  properties,  compendence  and 
perfectness,  are  independent  properties:  neither  of  them 
implies  the  other.  The  ensemble,  for  example,  of  the 
rational  points  of  a  straight  line  is  compedent,  but  it  is 
not  perfect  for  many  of  its  limit-points,— that  one,  for 
example,  whose  distance  from  the  origin  is  V2, — are  not 
members  of  the  ensemble.  On  the  other  hand,  the 
ensemble  composed  of  the  points,  (X  1,  2,  4  and  all  the  real 
points  between  0  and  1  and  all  between  2  and  4,  is  perfect, 
but  it  is  plainly  not  compendent  for  the  passage  from, 
say,  point  1  to  point  2  cannot  be  made  in  the  required 
way.  It  is  clear,  however,  that  the  system  or  ensemble 
of  the  real  numbers  is  at  once  perfect  and  compendent. 
The  same  is  true  of  the  segment  composed  of  zero,  1  and 
the  intervening  numbers;  it  is  true  of  the  ensemble  of 
points  of  a  straight  line  or  of  any  segment  of  it;  it  is  true 
of  the  ensemble  of  the  points  of  a  plane  or  of  the  ensemble 
composed  of  the  points  inside  and  of  those  on  the  cir- 
cumference of  a  circle;   and  so  on  and  on. 

These  considerations  may,  I  trust,  suffice  to  give  you  a 
general  notion  of  that  great  mathematical  instrument 
known  in  modern  analysis  as  the  Grand  Continuum  or 
as  the  mathematical  continuum  of  second  order  or  simply 
as  the  mathematical  continuum.  And  let  me  say  that 
you  will  miss  a  main  point  if  you  overlook  the  intimate 
connection  of  the  matter  with  the  Weber-Fechner  law. 

I  have  said  that  in  the  long  course  of  time  and  in  the 
interest  of  intellectual  harmony  or  freedom  the  various 
sense  "  continua  "  of  weights,  lengths,  sounds,  pressures, 
velocities  and  so  on,  have  been  gradually  replaced,  in  our 
thinking,  first  by  corresponding  conceptual  continua  of 
the  type  of  the  mathematical  continuum  of  first  order  and 


THE    MATHEMATICS    OF    PSYCHOLOGY      397 

second  by  conceptual  continua  of  the  type  of  the  mathe- 
matical continuum  of  second  order.  Of  course,  I  do  not 
mean  to  imply  that  the  first  replacement  was  completed 
before  the  second  began.  I  mean  merely  that  the  order 
indicated  is,  roughly  speaking,  correct.  Neither  do  I 
intend  to  imply  that  the  process  of  substitution  has  been 
always  a  conscious  one  nor  that  it  has  been  always  accom- 
panied by  a  realizing  sense  of  its  actuating  motive. 
Much  of  our  intellectual  life  is  not  attended  by  con- 
sciousness either  that  it  is  going  on  or  why  it  proceeds  in 
this  direction  rather  than  that.  That  the  replacements 
have  been  actually  made,  however,  is  sufficiently  evident 
in  the  fact  that  students  of  natural  science, — physicists, 
for  example,  or  astronomers  or  chemists, — habitually  and 
freely  employ  the  real  numbers,  whether  rational  or 
irrational,  algebraic  or  transcendental,  to  express  quan- 
tities or  amounts  of  the  various  kinds  of  physical  magni- 
tude. Are  these  students  aware  that  they  are  thus  deal- 
ing with  purely  conceptual  continua  and  not  with  such 
as  are  revealed  in  sense?  It  must  be  said  that,  for  the 
most  part,  they  are  not.  For  the  most  part  these  students 
are  not  indeed  aware  that  there  are  such  continua  even 
in  mathematics;  much  less  are  they  informed  regarding 
the  inner  structure  or  constitution  of  them;  they  employ 
them  naively,  as  children  may  handle  tools  which  they 
have  not  yet  analyzed.  This  is  not  said  in  any  spirit  of 
reproach  or  derogation,  for  it  is  only  in  very  recent  times 
that  even  mathematicians  themselves  have  made  the  con- 
stitution of  continua  a  subject  of  deliberate  investiga- 
tion, though  the  matter  figured  itself  vaguely  in  the  back- 
ground of  their  thought  for  more  than  two  thousand 
years.  Even  Aristotle  in  his  Physics  made  a  stab  at  the 
question. 


398  MATHEMATICAL    PHILOSOPHY 

Some  Questions. — Hereupon  certain  questions  natur- 
ally supervene.  One  of  them  is  this:  How  can  the 
symbols  or  elements  or  terms  that  constitute  the  mathe- 
matical continuum  be  effectively  employed  in  studying 
such  a  magnitude  as  pressure,  for  example,  or  gravity  or 
velocity?  The  answer  seems  to  reside  in  two  considera- 
tions: one  of  them,  which  I  have  hitherto  mentioned  in 
these  lectures,  is  the  fact  that  the  various  kinds  of  the 
magnitude  in  question  are  each  of  them  conceived  to  be 
composed  of,  or  decomposable  into,  parts  or  elements 
matching  in  a  one-to-one  way  the  elements  or  terms  that 
constitute  the  mathematical  continuum,  and  related 
among  themselves  as  the  terms  of  the  continuum  are 
related  among  themselves;  the  second  consideration  is 
the  fact  that  natural  science  is  concerned,  not  with  the 
constituents  of  a  magnitude,  but  only  with  the  relations 
among  them.  This  second  consideration,  which  so  easily 
escapes  attention,  is  one  of  those  fundamental  matters 
which  Professor  Poincare  never  wearied  of  insisting  upon. 
See  his  Science  and  Hypothesis,  for  example. 

Another  natural  question  is  this:  Does  the  replacement 
of  the  sensible  or  of  the  rational  continua  by  continua 
patterned  on  the  model  of  the  Grand  Continuum  guaran- 
tee us  against  all  difficulties  resembling  the  kind  repre- 
sented by  the  possibility  of  two  lines  crossing  without 
intersecting?  The  answer  is,  no.  This  particular  kind 
of  difficulty  has  indeed  been  overcome  by  means  of  the 
Grand  Continuum.  But  there  remain  to  surprise  us  other 
difficulties  of  a  somewhat  similar  kind.  Let  us  glance  at 
one  of  them.  Consider  a  sensible  curve  and  a  sensible 
straight  line.  We  can  always  dispose  them  so  that  they 
will  have  a  common  part  without  crossing.  Let  us  now 
replace  them,  in  thought,   by  corresponding  conceptual 


THE    MATHEMATICS    OF    PSYCHOLOGY      399 

magnitudes  of  the  type  of  the  second-order  mathematical 
continuum.  Next  let  the  breadth  and  thickness  diminish 
more  and  more,  keeping  always  a  common  part  without 
crossing.  It  appears  that,  at  the  limit,  the  common  part 
will  be  a  point  at  which,  however,  the  line  and  the  curve 
do  not  cross;  that  is  to  say,  it  appears  that  the  line 
becomes  a  tangent  to  the  curve  at  the  point.  In  some  such 
way,  it  came  to  be  believed  that  a  curve,  if  continuous 
at  any  point,  admits  a  tangent  at  the  point.  Nothing 
could  be  more  natural  and  the  belief  was  persistent  and 
long-lived.     Yet  we  know  to-day  that  in  such  matters 


Fig.  33. 

our  intuition,  precious  as  it  is,  cannot  be  implicitly 
trusted  for  we  know  today  that  the  belief  in  question  is 
false.  This  fact  may  be  shown  by  the  following  classical 
example.     Consider  the  locus  of  the  equation 

1 

y  =  x  sin  - 

J  x 

It  is  continuous  at  every  point  except  the  point  whose 
abscissa  is  #=0.  At  this  point  y  is  not  defined,  since 
division  by  zero  is  meaningless.  Hence  we  may  define 
it  as  we  please.  Let  us  agree  that  y  shall  be  zero  when 
x=0.  This  being  done,  the  curve  is  now  continuous  at 
the  point,  x  =  0,  as  well  as  at  all  other  points.  Differentiat- 
ing, we  get 

dy      •     I      1  1 

-~—  sin cos  - 

dx  x     x         x 


400  MATHEMATICAL   PHILOSOPHY 

for  the  slope  of  the  tangent  at  any  point  whose  abscissa 

dy 
x  yields  a  definite  value  for  -7-.     But  at  the  point,  x  =  0 

—  has  no  value;    it  is  indeed  meaningless  as  involving 

division  by  zero.  Hence  the  curve,  though  it  is  con- 
tinuous at  the  point  in  question,  does  not  admit  a  tangent 
at  the  point. 

As  you  may  know,  such  curves  are  infinitely  numerous. 
For  other  examples  one  may  consult  a  memoir  on  discon- 
tinuous functions  by  Darboux  in  the  Annates  de  I' '  Ecole 
Normale  superieure,  Vol.  IV,  2d  series.  Some  examples 
of  the  kind  are  beautifully  discussed  in  the  appendix 
of  W.  B.  Smith's  Infinitesimal  Analysis.  See  also  W.  K. 
Clifford's  remarks  on  "  Crinkly  Curves  "  in  Vol.  I  of  his 
Elements  of  Dynamic.  The  matter  is,  of  course,  dealt 
with  in  all  up-to-date  books  of  advanced  calculus  or  of 
the  theory  of  functions  of  the  real  variable.  By  a  famous 
example  adduced  by  Weierstrass,  it  was  shown  that  a 
curve  may  be  continuous  at  every  point  and  yet  have  no 
tangent  at  any  point. 

Phenomena  of  the  kind  above  indicated  naturally 
raise  another  question,  which  I  may  state  without  attempt- 
ing to  discuss  it  here.  The  question  is:  Is  it  possible  to 
construct  continua  of  higher  than  second  order,  continua, 
that  is,  whose  elements  are,  so  to  speak,  compacted 
more  closely  together  than  in  the  case  of  the  Grand 
Continuum;  and  what  would  be  the  bearing  of  such  a 
higher  continuum  upon  the  sort  of  phenomena  above 
indicated?  The  first  part  of  the  question  is  discussed  by 
Paul  Du  Bois-Reymond.  The  works  of  this  author  are 
not  easy  to  read.  The  student  may  be  referred  to  G.  H. 
Hardy's  Orders  of  Infinity:    the  InHnitar-calciil  of  Paid 


THE    MATHEMATICS    OF    PSYCHOLOGY      401 

Du  Bois-Reymond,  where  the  chief  ideas  of  the  latter 
are  presented  and  his  works  cited. 

Conquest  or  Transcendence  of  Sensibility  Thresholds. — 
By  way  of  emphasis, — for  the  matter  is  very  important, 
especially  for  psychology, — let  us  state  explicitly  that  the 
conceptual  continua  which  we  have  been  discussing  at  so 
great  length  are  not  infected,  as  are  sense  "  continua," 
with  the  presence  of  thresholds.  They  are  free  from 
initial  thresholds,  for  any  such  conceptual  continuum 
is  composed  of  parts  that  continuously  decrease  in  size 
down  to  zero.  They  are  free  from  terminal  thresholds 
because  they  each  of  them  present  a  sequence  of  parts 
increasing  beyond  any  assigned  finite  amount,  however 
great.  They  are  free  from  difference  thresholds,  for  any 
difference,  however  small,  between  portions  of  a  con- 
ceptual continuum  is  conceptually  discernible.  In  a 
sense  "  continuum  "  there  are,  properly  speaking,  neither 
infinitesimals  nor  infinites,  but  in  a  conceptual  con- 
tinuum there  are  both.  In  a  word  we  may  say  that 
conception,  or  thought,  is  a  kind  of  infinitely  refined  sensi- 
bility, for  there  is  no  quantity  too  small  for  thought  to 
detect  and  to  discriminate  from  any  other.  Here  this 
lecture  must  close.  Our  topic  has  been  the  mathematics  of 
psychology;  the  discussion,  you  see,  has  inevitably  led 
us  pretty  far  into  our  next  topic — the  psychology  of 
mathematics — and  even  into  the  psychology  of  science 
in  general. 


LECTURE  XIX 
The  Psychology  of  Mathematics 

RETARDATION  OF  MATHEMATICS  AND  SCIENCE  BY  BACK- 
WARD PSYCHOLOGY PSYCHOLOGY  OF  MATHEMATICS 

ESSENTIAL    TO    BEST    MATHEMATICAL    TEACHING 

QUESTIONS       FOR      PSYCHOLOGISTS SYMMETRY      OF 

THOUGHT  AND  ASYMMETRY  OF  IMAGINATION. 

In  the  preceding  lecture  we  saw  that  a  little  study  in 
the  "mathematics  of  psychology"  led  us  quickly  and 
naturally  into  the  "psychology  of  mathematics"  and  even 
into  that  of  science  in  general.  Yet  we  are  now  going  to 
discuss  the  "psychology  of  mathematics"  as  if  its  field 
were  well  defined  and  did  not  run  into  all  other  subjects, 
which,  in  their  turn,  penetrate  it.  Nature  does  not  greatly 
respect  our  little  academic  custom  of  carving  her  up  and 
calling  the  pieces  departments  of  study.  Chemistry, 
physics,  mechanics,  geometry,  metaphysics,  psychology, 
logic,  ethics,  esthetics,  and  the  rest  all  penetrate  and 
overflow  the  walls  we  surround  them  with,  and  mingle 
their  waters  in  one  vast  sea.  In  the  great  world  of 
Nature, — the  subject  of  all  thought, — there  are  emphases 
indeed  but  no  fixed  divisions  corresponding  to  our  pretty 
ologies,  ographies,  and  ics,  and  even  the  emphases  per- 
petually shift  their  incidence.  And  yet  there  is  a  sense 
in  which  such  divisions  and  walls  are  not  artificial  but  are 

402 


THE    PSYCHOLOGY    OF    MATHEMATICS      403 

natural  for  they  are  made  by  man,  and  man  is  a  part  of 
nature — the  part  that  studies  the  whole.  In  that  sense, 
man-made  divisions  of  nature  are  natural  divisions,  made 
by  Nature  herself  to  facilitate  the  process  of  self-under- 
standing; but  they  are  not  aboriginal  and  they  are  not 
permanent, — they  are  experimental  devices,  mere  con- 
veniences for  the  service  of  a  people  or  an  age,  and  des- 
tined to  change. 

There  was  a  time  when  what  we  now  call  logic  and 
what  we  now  call  psychology  were  not  distinguished — 
not  held  apart;  today  they  are;  and  so,  given  any  scien- 
tific or  philosophic  subject,  we  habitually  speak  of  its 
logic  and  its  psychology;  the  two  things,  though  the  sub- 
ject is  one,  represent  different  types  of  interest  in  it,  dif- 
ferent emphases  or  aspects  of  it.  Accordingly,  mathe- 
matical science  presents  two  fields  of  interest:  the  logic 
of  mathematics  and  the  psychology  of  mathematics.  In 
the  former,  research  has  achieved  great  results;  in  the 
latter,  but  little  of  solid  worth.  Why  the  great  disparity? 
Because  the  logic  of  mathematics  has  interested  philo- 
sophic-minded logicians  who  were  at  the  same  time  mathe- 
maticians; while  those  who  have  dealt  with  the  psychology 
of  mathematics  have  not  known  mathematics  well  enough 
to  know  what  it  was  they  were  attempting  to  psychologize 
or  even  to  know  that  they  did  not  know — witness,  for 
example,  the  literature  of  the  "psychology  of  number." 
In  the  introductory  lecture  I  pointed  out  that  modern 
research  in  the  logic  of  mathematics  has  culminated  in 
a  really  marvelous  thesis:  rightly  understood,  mathematics 
and  logic  are  identical, — the  two  are  one  science, — logic 
(in  its  tradition  sense)  being  the  earlier  part  of  that 
science,  and  mathematics  (in  its  traditional  sense)  being 
its  later  part.     We  could,  therefore,  say  with  perfect  jus- 


404  MATHEMATICAL    PHILOSOPHY 

tice  that  what  we  arc  now  to  discuss  is  the  psychology  of 
logic.  For  our  present  purpose  it  is  better,  however, 
to  say  psychology  of  mathematics;  the  other  title  is  too 
unfamiliar. 

It  is  my  aim  to  signalize  the  importance  of  the  sub- 
ject, to  suggest  a  few  of  its  problems,  and  thus  possibly 
to  incite  some  mathematician  to  acquire  sufficient  psy- 
chological competence,  or  some  psychologist  to  acquire 
sufficient  mathematical  competence,  to  deal  with  it  effect- 
ively. The  subject  no  doubt  possesses  great  interest  in 
itself.  "The  genesis  of  mathematical  discovery,"  says 
Poincare  in  Science  and  Method,  "is  a  problem  which 
must  inspire  the  psychologist  with  the  keenest  interest." 
But  that  is  not  the  main  point.  The  main  point  is  that 
the  general  neglect  of  the  subject  by  competent  men 
throughout  the  centuries  has  greatly  retarded  the  prog- 
ress of  science;  it  has,  in  the  first  place,  retarded  the 
progress  of  mathematics  and  it  has  thus  retarded  the 
progress  of  all  the  sciences  whose  prosperity  depends 
upon  that  of  mathematics.  That  such  is  the  case,  a  few 
considerations  will  make  sufficiently  evident. 

Consider,  for  example,  the  birth  and  development  of 
the  concept  of  hyperspaces,  which,  as  we  saw  in  a 
previous  lecture,  is  less  than  a  hundred  years  old. 
We  have  noted  its  great  importance  for  mathematics, 
for  physical  science  and  for  philosophy.  Why  was 
the  advent  of  this  great  concept  so  long  delayed?  The 
answer  is:  because  the  psychology  of  mathematics  was 
so  little  understood.  For  many  centuries  the  concept  in 
question  had  been  knocking  at  the  door  but  it  was  not 
admitted  because  psychologically  ignorant  mathematicians 
and  psychologically  ignorant  philosophers  believed  that 
mathematical   concepts,    if   they  be   not  indeed   concepts 


THE    PSYCHOLOGY    OF    MATHEMATICS      405 

of  sensible  or  perceptible  things,  must  at  all  events  be 
concepts  of  things  that  we  can  imagine.  The  evidence 
supporting  my  answer  is  unmistakable  and  conclusive.  Let 
us  consider  some  of  it.  I  am  here  greatly  indebted  to 
Professor  Manning's  admirable  "Introduction"  to  his 
Geometry  of  Four  Dimensions.  In  the  following  quota- 
tions from  various  authors  cited  by  him,  I  shall  take  the 
liberty,  for  it  will  be  helpful,  to  italicise  some  of  their 
words. 

In  the  De  Caelo  of  Aristotle  we  are  told  that  "The 
line  has  magnitude  in  one  way,  the  plane  in  two  ways, 
and  the  solid  in  three  ways,  and  beyond  these  there  is  no 
other  magnitude  because  the  three  are  ally  It  is  plain 
that  the  "all"  is  an  "all"  for  imagination,  not  for  con- 
ception. And  we  are  further  told  that  "There  is  no 
transfer  into  another  kind,  like  the  transfer  from  length 
to  area  and  from  area  to  solid."  The  statement  is  true 
for  perception  and  for  imagination;  but  for  thought  or 
conception,  it  is  false. 

For  another  instance  in  point  consider  the  following 
statement  (of  the  sixth  century,  A.  D.)  found  in  the 
Commentaries  of  Simplicius:  "The  admirable  Ptolemy 
in  his  book  On  Distance  well  proved  that  there  are  no 
more  than  three  dimensions,  because  of  the  necessity  that 
distances  should  be  defined,  and  that  the  distances  defined 
should  be  taken  along  perpendicular  lines,  and  because 
it  is  possible  to  take  only  three  lines  that  are  mutually  per- 
pendicular, two  by  which  the  plane  is  defined  and  a  third 
measuring  depth;  so  that  if  there  were  any  other  distance 
after  the  third  it  would  be  entirely  without  measure  and 
zvithout  definition.  Thus  Aristotle  seemed  to  conclude 
from  induction  that  there  is  no  transfer  into  another 
magnitude,   but  Ptolemy  proved   it."      Here  it   is  again 


406  MATHEMATICAL   PHILOSOPHY 

evident  that  the  psychology  of  mathematics  had  not  yet 
learned  to  discriminate  the  conceivable  from  the  imagin- 
able. We  now  know  that  in  a  space  of  n  dimensions  n 
lines  can  be  mutually  perpendicular  and  we  know  that 
such  spaces  are  geometrically  just  as  good  as  any  other. 
And  not  only  was  the  progress  of  geometry  thus  retarded 
but  that  of  algebra,  too,  in  as  great  or  greater  measure; 
for  not  only  were  lines,  surfaces  and  solids  things  in 
imagination's  realm  but  so,  too,  were  numbers;  it  is  well 
known  that  for  the  Greek  mathematicians  and  a  long 
series  of  their  successors,  numbers  were  geometric  things 
— one  number  was  a  line  (segment),  a  product  of  two 
numbers  was  a  rectangle  or  a  square,  and  that  of  three 
a  parallelopiped  or  a  cube.  False  psychology  thus 
balked  the  advancement  of  equation  theory,  for  x3  was 
indeed  a  real  thing, — a  geometric  cube, — but  what,  pray, 
was  #4,  for  example,  or  .v5,  and  so  on?  The  answer  was 
evident — they  were  unreal.  So,  in  the  sixteenth  century, 
we  are  told  by  Stifel,  reviser  of  Rudolph's  Algebra,  that 
"going  beyond  the  cube  just  as  if  there  were  more  than 
three  dimensions"  is  a  thing  "against  nature."  And  in 
the  following  century  John  Wallis  regards  the  giving  of 
"ungeometrical"  names  to  the  fourth  and  higher  powers 
of  numbers  as  quite  intolerable.  They  are,  he  says,  a 
"Monster  in  Nature,  less  possible  than  a  Chimaera  or 
Centaure."  Why?  Because  "Length,  Breadth  and 
Thickness  take  up  the  whole  of  Space.  Nor  can  Fansie 
imagine  how  there  should  be  a  Fourth  Local  Dimension 
beyond  these  Three."  Here  note  again  what  the  barrier 
is, — a  childishly  naive  psychology, — Nature  does  not 
transcend  the  imaginable, — what  is  merely  conceivable  is 
monstrous, — a  psychology  so  rude  and  so  crude  that,  were 


THE    PSYCHOLOGY    OF    MATHEMATICS       407 

it  strictly  applied,  the  very  possibility  of  mathematics, 
rightly  understood,  would  be  thereby  excluded. 

It  would  not  be  difficult  to  produce  much  more  evi- 
dence of  similar  kind.  But  I  will  content  myself  with 
one  further  citation — one  that  is  less  than  a  century  old 
and  comes  from  a  mathematician  of  great  power.  I  re- 
fer to  Mobius,  author  of  Der  barycentrische  Calcul 
(1827).  He  saw  indeed  that,  if  there  were  a  space  of 
four  dimensions,  it  would  be  possible  to  rotate  in  it  a  solid 
figure  of  ordinary  space  just  as  a  plane  figure  can  be 
rotated  in  the  latter  space;  and  he  saw  that,  if  such  a 
rotation  of  solids  were  possible,  we  could  make  two  sym- 
metric solids  coincide  just  as  we  can  make  two  symmetric 
plane  figures  coincide  by  rotation  in  space  immersing  the 
plane.  This  is  perfectly  good  mathematics,  which,  how- 
ever, he  rejects  because  of  a  false  psychology.  His 
statement  is  this:  "Da  aber  ein  soldier  Raum  nicht 
gedacht  werden  kann,  so  ist  auch  die  Coincidenz  in  diesem 
Falle  unmbglich"  He  meant  that  such  a  space  cannot 
be  imagined — he  could  not  have  meant  that  it  cannot  be 
conceived,  for  he  had  already  conceived  it;  his  blunder 
was  not  one  in  logic;  it  was  a  blunder  in  psychology — the 
psychology  of  mathematics;  though  an  able  mathemati- 
cian, he  did  not  know  that  a  conceivable  space  and  a  con- 
ceivable rotation  are  perfectly  good  mathematically,  even 
though  they  transcend  the  domain  of  imagination. 

The  foregoing  facts  show  clearly  that  a  backward 
psychology  of  mathematics  not  only  operated  to  hamper 
the  progress  of  algebra,  but  actually  delayed,  for  more 
than  two  thousand  years,  the  advent  of  the  concept  of 
hyperspace  and  w-dimensional  geometry. 

If  you  turn  to  the  genesis  of  non-Euclidean  geometry, 
you  find  an  essentially  similar  tale.     The  birth  was  baf- 


408  MATHEMATICAL    PHILOSOPHY 

fled  for  over  twenty  centuries.  Baffled  by  what?  By  a 
psychology  which  recognized  no  space  except  our  sensuous 
space,  which  believed  that  our  sensous  space  is  geome- 
trizable,  that  Euclid's  axioms  are  "self-evident"  truths 
regarding  it,  and  that  his  Elements  embodies  an  exact 
description  of  it;  by  a  psychology  which,  therefore,  could 
not  contemplate  even  the  possibility  of  wow-Euclidean 
geometry  and  which,  when  such  a  geometry  was  at  length 
devised  in  spite  of  it,  insisted  upon  trying  its  claims  in  the 
courts  of  sensibility  and  perception  and  imagination,  not 
knowing  that,  in  questions  regarding  the  logical  validity 
of  geometric  science,  those  courts  are  entirely  without 
jurisdiction. 

Let  me  allude  briefly  to  another  branch  of  modern 
mathematics — projective  geometry.  It  was  invented,  we 
have  seen,  in  the  seventeenth  century,  lost,  forgotten,  and 
re-invented  in  the  nineteenth.  Why  not  before — centuries 
before?  What  was  in  the  way?  Logic?  Not  primarily; 
it  was  psychology — a  false  psychology  of  mathematics. 
The  invention,  as  you  know,  required  the  conception  of 
infinitely  distant  points  and  the  conception  of  lines  and 
planes  such  that,  if  parallel,  they  meet  in  those  points.  But 
how  could  that  happen?  How  could  parallels  meet? 
They  could  not,  it  was  said, — it  was  psychologically  im- 
possible,— the  possibility  was  denied  by  sense,  denied  by 
perception,  and,  most  conclusive  of  all,  denied  by  imagi- 
nation. 

I  have  just  now  mentioned  "infinitely"  distant  points. 
We  are  thus  reminded  of  the  modern  concept  of  infinity, 
— of  infinite  classes,  ensembles,  sets,  or  manifolds, — the 
subject  of  Lecture  XV.  We  have  seen  that  this  great 
concept,  though  it  is  classic  today,  was  born  but  yesterday. 
Why  not  a  thousand  or  two  thousand  years  ago?     Well, 


THE    PSYCHOLOGY    OF    MATHEMATICS      409 

it  was  born  or  well-nigh  born,  as  we  have  seen,  to  the 
genius  of  Epicurus  and  had  the  fortune  to  inspire  the 
great  poem  of  Lucretius.  With  this  exception,  it  had 
no  career  in  science  and  none  in  philosophy — it  was 
sterile.  Why?  The  same  old  trouble — a  shallow  psy- 
chology. For,  as  you  know,  an  infinite  class  must  have 
a  part  containing  as  many  things  as  the  whole  class  con- 
tains. But  who  ever  saw  such  a  class  or  ever  imagined 
one?  "Nonsense!"  exclaimed  psychology,  and  the  great 
conception, — so  important  for  science,  for  philosophy 
and  for  rational  theology, — slumbered  for  twenty  cen- 
turies. 

Passing  to  another  field,  we  find  that  the  development 
or  generalization  of  the  number  concept  was  greatly 
hampered  by  the  same  cause.  The  descriptive  terms, — 
"surd"  (which  means  absurd),  "irrational,"  "imaginary," 
and  "impossible," — which  were  applied  to  large  classes 
of  numbers  that  had  been  literally  forced  upon  the  atten- 
tion of  mathematicians  by  familiar  operations,  sufficiently 
tell  the  tale.  Mathematically  those  numbers,  as  we  now 
know,  were  quite  as  genuine,  quite  as  legitimate,  as  the 
ordinary  integers  and  fractions.  Why,  then,  were  they 
called  "surd,"  "irrational,"  "imaginary,"  and  "impos- 
sible"? Because  they  encountered  a  psychology  that  did 
not  understand  the  nature, — the  mental  nature, — of 
mathematical  generalization:  a  psychology  which  held 
that  the  new  "numbers,"  in  order  to  be  legitimate,  must 
conform  to  the  familiar  laws  of  the  old  ones  and  must, 
moreover,  like  the  old  ones,  admit  of  interpretation  or 
application  in  the  so-called  "actual"  world  of  sense-per- 
ception. 

The  history  of  many  another  mathematical  develop- 
ment bears  similar  witness.     But  we  need  not  pursue  the 


410  MATHEMATICAL   PHILOSOPHY 

matter  further.  The  evidence  now  before  us  is  sufficient. 
It  is  perfectly  clear  that  in  course  of  the  centuries  the 
progress  of  mathematics  has  been  much  retarded,  some- 
times arrested  for  long  periods  or  diverted  from  its 
natural  course,  by  a  psychology  which,  in  things  mathe- 
matical, often  did  not  know  a  knee  from  an  elbow. 

Is  mathematics  retarded  by  that  cause  today?  I 
believe  that  mathematical  research  is  not  much  thus  re- 
tarded— at  all  events  not  directly.  No  doubt  the  number 
of  mathematicians  who  are  also  expert  psychologists  is 
very  small.  But  research  mathematicians  usually,  though 
not  always,  understand  the  psychology  of  their  own  science 
well  enough  to  recognize  a  mathematical  idea  as  being 
such,  wherever  and  whenever  it  occurs.  If  it  be  a  new 
one  and  be  found  to  be  interpretable  in  the  world  of 
perception  or  in  the  world  of  imagination,  they  are 
thereby  rejoiced,  naturally  so;  but  if  it  be  not  thus  in- 
terpretable, as  it  may  not  be,  they  are  not  so  psychologic- 
ally unenlightened  as  to  refuse  it  hospitality  on  that 
account.  The  history  of  their  subject  has  taught  them 
better. 

But  mathematical  research  and  the  dissemination  of 
mathematical  knowledge  are  very  different  things.  In 
respect  of  the  latter,  I  have  no  doubt  that,  if  teachers 
of  mathematics  were  better  trained  in  psychology  and 
especially  in  the  psychology  of  mathematics,  their  teach- 
ing would  be  far  more  effective;  for  questions  of  logic 
would  then  be  seen  and  set  in  clearer  light,  less  frequently 
confused  with  psychological  considerations,  while  the 
latter,  presented  as  such,  would  often  contribute  to  the 
instruction  a  light  of  their  own.  Will  you  allow  me  a 
word  of  personal  experience?  I  count  it  a  great  good 
personal  fortune  that  as  a  young  man  I  received  mathe- 


THE    PSYCHOLOGY    OF    MATHEMATICS      411 

matical  instruction  from  one  in  whose  teaching  the  logic, 
the  philosophy,  the  psychology,  and  the  poetry  of  the 
subject  mingled  together  and  fortified  each  other  like  the 
parts  of  an  orchestra.  I  refer  to  Professor  William 
Benjamin  Smith,  now  of  world-wide  fame  as  a  Biblical 
scholar  and  critic.  I  am  not  going  to  enlarge  here  upon 
this  important  matter  of  making  the  psychology  of 
mathematics  effective  in  mathematical  instruction,  but  will 
merely  refer  you,  for  some  relevant  suggestions,  to  the 
earlier  lectures  where  distinctions  of  logical  and  psycho- 
logical were  repeatedly  indicated  and  where,  especially 
in  Lecture  VII,  in  connection  with  the  psychological  dis- 
crimination of  logically  identical  doctrines,  was  intro- 
duced the  important  notion  of  "excessive  meaning." 

We  have  been  talking  about  the  neglect  and  back- 
wardness of  the  psychology  of  mathematics.  Thus  far 
we  have  referred  mainly  to  the  neglect  of  it  by  mathe« 
maticians.  What  are  we  to  say  of  its  neglect  by  profes- 
sional psychologists?  I  have  no  desire  to  be  fault-finding, 
querulous  or  unjust.  I  am  well  aware  that  psychologists 
have  many  things  to  occupy  their  attention — that  their 
field  is  vast,  diversified  and  complicate.  I  know  that, 
like  other  scientific  folk,  they  are  obliged  to  select.  I 
know  that  for  an  outsider  to  attempt  to  dictate  or  pre- 
scribe their  choice  would  be  presumptuous.  At  the  risk, 
however,  of  seeming  impertinent, — which  usually  means 
a  little  too  pertinent, — I  venture,  as  an  interested  layman, 
to  suggest  that,  in  neglecting  the  psychology  of  mathe- 
matics, professional  psychologists  not  only  neglect  an 
obligation  to  mathematics  and  natural  science  but  also 
neglect  an  exceedingly  interesting  subdivision  of  their 
own  proper  field.  For  their  field  is  Mind, — psychology, 
we  are  told,  is  the  study  of  mind,  the  study  of  mental 


412  MATHEMATICAL   PHILOSOPHY 

phenomena, — and  I  believe  we  may  assume  that,  where 
there  is  mathematics,  there  is  some  manifestation  of  mind, 
— that  mathematics,  regarded  as  an  enterprise,  is  an 
enterprise  of  mind, — that,  regarded  as  a  body  of  achieve- 
ments, it  is  a  body  of  mental  achievements, — that,  re- 
garded as  a  mode  of  life,  it  is  a  mode  of  mental  life, 
— that,  in  a  word,  mathematical  phenomena  represent 
mental  phenomena  and  are  unsurpassed  as  means  in  the 
study  of  mind.  I  do  not  mean  that  all  kinds  of  mental 
phenomena  are  thus  represented.  Lust,  for  example,  is 
not,  nor  fear,  nor  anger,  nor  hate,  nor  malice,  nor  envy, 
nor  many  another  such  amiable  propensity  of  unregen- 
erate  souls — of  course  I  am  speaking  here  of  mathe- 
matics and  not  of  mathematicians,  who  have  many  inter- 
esting qualities  that  their  science  has  not.  But  perception 
—  discrimination  —  imagination  —  fantasie  —  concep- 
tion —  judgment  —  analysis  —  synthesis  —  reasoning  — 
generalization  —  the  energy  of  will  —  the  restraint  of 
passion — the  sensibility  and  daring  of  genius — the  sense 
for  order,  for  symmetry,  for  harmony,  for  intellectual 
beauty,  for  cogency  and  clarity  of  thought, — where  out- 
side of  mathematics  do  such  mental  phenomena,  which  it 
is  the  psychologist's  profession  to  examine,  show  them- 
selves in  so  clear  a  light?  It  is  indeed  obvious  that  the 
whole  literature  of  mathematics  may  be  read  and  in- 
terpreted as  a  commentary  upon  the  nature  of  the 
human  mind.  Select,  for  example,  a  well-wrought  dem- 
onstration and  examine  it.  What  can  you  say  of  it? 
You  can  say  this:  A  normal  human  mind  is  such  that, 
if  it  begin  with  such-and-such  principles  or  premises  and 
with  such-and-such  ideas  and  if  it  combine  them  in  such- 
and-such  ways,  moving  from  step  to  step  in  such-and-such 
an  order,  it  will  find  that  it  has  thus  passed  from  dark- 


THE    PSYCHOLOGY    OF    MATHEMATICS      413 

ness  to  light, — from  doubt  to  conviction.  Obviously  such 
a  proposition  is  not  mathematical;  it  is  psychological — 
it  states  a  fact  respecting  the  nature  of  a  normal  human 
mind.  Such  interpretations  of  mathematical  literature 
are  psychologically  very  illuminating;  the  possibility  of 
making  them  is  so  evident,  once  it  is  pointed  out,  that  I 
should  have  refrained  from  mentioning  it  except  for  the 
fact  of  its  being  commonly  overlooked  and  neglected. 

For  another  example,  consider  the  phenomenon  of 
generalization, — the  process  by  which  the  human  mind 
from  time  to  time  enlarges  the  empire  of  its  rational 
activity.  What  is  generalization  as  a  process  of  mind, 
as  a  mental  event?  What  are  the  mental  phenomena 
involved?  How?  In  what  relations?  I  am  not  going 
to  attempt  to  answer  here.  I  wish  merely  to  propose  the 
problem  to  students  of  psychology.  Generalization  occurs 
in  all  fields  of  thought  but  in  mathematics  it  may  be  seen 
in  its  nakedness.  There,  then,  is  the  best  place  to  study 
it  as  a  phenomenon  of  mind.  Take,  for  example,  the 
striking  succession  of  generalizations  by  which  the  domain 
of  the  number  concept,  which  once  contained  nothing  but 
our  familiar  integers,  has  been  gradually  extended  to 
embrace  positives  and  negatives,  rationals  and  irrationals, 
reals  and  imaginaries,  cardinals  and  ordinals,  including 
the  transfinite  numbers  of  Georg  Cantor;  or  take  the 
no  less  striking  series  of  generalizations  by  which  the 
conception  of  geometry  has  been  enlarged.  As  specimens 
of  generalization,  those  alluded  to  are  probably  the  best 
to  be  found  in  the  history  of  thought.  I  venture  to  com- 
mend them  as  such  to  students  of  mind.  Some  of  you 
are  psychologists.  If  you  will  study  the  great  process 
of  generalization  by  help  of  the  specimens  mentioned, 
then  the  rest  of  us  will  go  to  you  confidently, — as  laymen 


414  MATHEMATICAL   PHILOSOPHY 

to  experts, — for  enlightenment.  For  there  are  questions 
to  be  asked.  Generalization  seems  to  be  sometimes  very 
simple  and  sometimes  very  complicate.  We  should  like 
to  know  what  mental  phenomena, — what  sorts  of  mental 
activity, — are  involved  in  it.  What,  if  any,  is  the  role 
of  imagination  in  it,  and  that  of  conception  and  that  of 
reasoning?  Does  generalization  transcend  the  realm  of 
imagination?  What  is  the  office  of  logic  therein?  Is 
generalization  the  end  of  a  series  of  operations  or  is  it 
the  beginning  of  a  new  series?  Is  it  a  conclusion  forced 
by  reason  or  does  it  involve  a  creative  act  of  will  stimu- 
lated by  motives  but  not  coerced  by  them?  What  are 
the  actuating  motives  of  the  process?  Are  all  general- 
izations essentially  alike?  If  not,  what  are  the  kinds, 
and  how  do  they  differ?  How  do  the  phenomena  of 
scientific  generalization  compare  with  those  of  idealiza- 
tion in  other  fields?  Such  questions  are  neither  primarily 
mathematical  nor  primarily  metaphysical;  they  are  psy- 
chological questions,  which  it  is  your  proper  function  as 
students  of  mind  to  investigate  for  your  own  enlighten- 
ment and  for  that  of  others.  Let  me  cite  again  the 
statement  of  that  great  man,  Henri  Poincare:  "The 
genesis  of  mathematical  discovery  is  a  problem  which 
must  inspire  the  psychologist  with  the  keenest  interest." 
The  things  I  have  been  saying  are  submitted  as  sug- 
gestions only;  being  a  layman's  suggestions,  they  are 
probably  very  inferior  to  the  best  that  could  be  made. 
I  am  tempted,  nevertheless,  to  add  yet  another  one.  It 
is  that  a  good  way, — perhaps  the  best  way, — for  psy- 
chologists to  advance  their  own  subject  would  be  to 
cooperate  with  philosophic  mathematicians  and  philo- 
sophic physicists  in  their  efforts  to  solve  the  great  problem 
mentioned  near  the  close  of  Lecture  X, — the  problem,  I 


THE    PSYCHOLOGY    OF    MATHEMATICS      415 

mean,  of  discovering  the  relations  between  the  data  of 
sense  and  the  conceptual  objects  of  science, — the  problem, 
in  other  words,  of  ascertaining  whether,  how,  and  to 
what  extent  such  conceptual  and  hypothetical  objects 
(points,  instants,  space,  time,  atoms,  electrons,  ether, 
etc.)  can  be  replaced  by  objects  actually  constructed  out 
of  sense-given  data  and  having  the  properties  demanded 
by  science.  If  such  constructions  can  be  made,  science 
will  be  able  to  dispense  with  many  hypotheses  or  many 
"as  ifs."  That  problem,  it  is  evident,  is  a  truly  great 
one. 

In  this  lecture  (as  also  in  preceding  ones)  I  have 
repeatedly  emphasized  the  importance  of  a  certain  psy- 
chological distinction  which  I  have  called  the  distinction 
between  imagination  and  conception.  It  is  today  well 
recognized  by  all  mathematicians.  They  are  accustomed 
to  designating  it, — not  quite  happily,  I  believe, — as  the 
distinction  between  "intuition"  and  "analysis."  It  is  the 
distinction  between  the  power  of  the  mind  to  picture 
and  its  power  to  think.  We  have  seen  that  failure  to 
make  it  has  often  retarded  the  progress  of  mathematics 
directly  and  that  of  kindred  sciences  indirectly.  It  is 
absolutely  essential  to  the  philosophy  of  science;  without 
it  the  history  of  thought  cannot  be  understood.  I  am 
here  reminding  you  of  the  matter  because  the  considera- 
tions with  which  I  intend  to  close  will  incidentally  shed 
new  light  upon  it. 

I  wish  to  call  your  attention  to  certain  contrasting 
psychological  phenomena  that  seem  not  to  have  found 
recognition  in  the  literature  of  psychology.  I  shall  pre- 
sent them  without  attempting  to  explain  them.  What  I 
wish  to  point  out  is  that,  in  relation  to  space,  conception 
or  thought  is  symmetric  in  its  representations  and  that 


416  MATHEMATICAL    PHILOSOPHY 

imagination   is   not.      My   theme   is :    The   symmetry   of 
thought  and  the  asymmetry  of  imagination.1 
Consider  the  simple  algebraic  expression 

The  u's  precede  the  x's,  but  that  is  here  of  no  importance, 
for,  owing  to  the  commutative  law  of  ordinary  multiplica- 
tion, ux  is  equivalent  to  xu;  if  we  replace  the  u's  by  the 
corresponding  x's  and  the  latter  by  the  former,  the 
expression  remains  algebraically  unaltered.  On  that  ac- 
count we  say  that  the  expression  is  symmetric  with  respect 
to  the  us  and  the  x's.  Such  interchange  of  the  us 
and  the  x's  may  be  likened  to  the  interchange  of  two 
opposite  halves  of  a  perfectly  symmetric  tree — the  figure 
of  the  tree  as  a  whole  remains  unchanged.  It  will  be 
convenient  to  denote  the  expression  by  the  symbol 
E{u>  x) — the  symbol  E(x,  u)  would,  of  course,  do  just 
as  well  but  let  us  use  the  former. 
Now  consider  the  equation 

(i)  E{u,  x)  =0 

It  is,  like  the  expression,  symmetric  in  the  sense  defined. 
We  may  interpret  the  equation  geometrically.  To  do  so, 
let  us  view  the  x's  as  coordinates  of  a  point  in  a  point- 
space,  Sn,  of  n  dimensions.  If  we  suppose  the  u's  to  have 
definite  values  the  equation  (i)  imposes  one  condition 
on  the  mobility  of  the  point  {x\,  X2,  .  .  .  ,  xn);  and  so  the 
equation  represents, — has  for  its  locus,  as  we  say, — a 
space  S„_x  of  points.  If  we  give  the  u's  another  set 
of  values,  thus  obtaining  a  new  equation  of  form   (i), 

JA  paper  on  this  subject  which  I  presented  at  the  Princeton  meeting 
of  the  American  Philosophical  Association  (1910)  was  published  in  the 
Journal  of  Philosophy,  Psychology  and  Scientific  Method,  June  8,  191 1. 


THE    PSYCHOLOGY    OF    MATHEMATICS      417 

the  new  equation  will  represent  another  space  of  n  —  I 
dimensions.  It  is  thus  plain  that  the  u's  serve  for  coor- 
dinates for  a  variable  Sn_1  in  Sn  just  as  the  x's  serve  for 
coordinates  of  a  point  in  Sa.  Since  the  us  may  take  as 
many  different  systems  of  values  as  the  x's  may  take,  you 
see  that  the  space  S„,  in  which  we  are  operating,  contains 
as  many  S„_x's  as  it  contains  points. 

We  have  just  now  seen  that,  if  the  us  be  held  fixed 
in  value  and  the  x's  be  allowed  to  vary  subject  to  condi- 
tion (i),  this  equation  represents  some  definite  Sn_1  as 
the  ensemble  or  locus  of  the  points  contained  in  it.  Now 
note  very  carefully  the  reciprocal  or  dual,  as  it  is  called, 
of  the  fact  just  stated.  The  dual  is  that,  if  the  x's  be  held 
fixed  in  value  (thus  giving  us  a  fixed  point,  say,  P)  and 
the  u's  be  allowed  to  vary  subject  to  condition  (i),  the 
equation  represents  P  as  the  ensemble  or  envelope  (as  it 
is  called)  of  all  the  Sn_^s  containing  it. 

Naturally  the  two  interpretations — one  for  the  us 
fixed  and  the  x's  variable,  the  other  for  the  x's  fixed  and  the 
u's  variable — of  one  and  the  same  equation 

E(u,  x)  =0=E{x>  u) 

may  be  significantly  described  as  symmetric  interpreta- 
tions. Indeed,  as  you  readily  see,  if  in  the  conceptual 
space  Sn  (of  operation)  we  interchange  the  notion  of 
point  (as  an  envelope  of  Sn_^s)  and  the  notion  of  Sn_1 
(as  a  locus  of  points),  Sn  will  as  a  whole  remain,  like  the 
initial  expression,  like  our  equation,  like  our  symmetric 
tree,  absolutely  unchanged.  Under  the  mentioned  opera- 
tion, Sn  is  an  invariant.  In  the  same  way,  systems  of 
equations  like  the  foregoing  one  admit  of  symmetric 
interpretations.  But  I  shall  not  deal  with  such  systems 
where  n  is  general.     It  will  be  easier  for  you,  and  for  my 


418  MATHEMATICAL   PHILOSOPHY 

purpose  it  will  be  sufficient,  to  begin  with  the  simple  case 
where  n  =  2  and  to  observe  what  happens  when  n  is  taken 
larger  and  larger. 

It  is  essential  to  note  the  fact  that  the  above-given 
symmetric  interpretations  are  conceptual — interpretations 
by,  in  and  for  pure  thought.  It  is  equally  essential  to 
note  that  our  "  spatial  "  imagination  or  intuition  or 
picturing  power  attempts  to  imitate  them, — attempts 
to  make  in  its  way  parallel  interpretations, — interpreta- 
tions, that  is,  which  correspond  to  or  match  in  detailed 
one-to-one  fashion  the  thought  interpretations.  In  other 
words,  imagination  endeavors  to  find  in  its  own  domain 
images,  pictures  or  objects  to  match  the  conceptual 
objects — points,  Sn_1's,  lines,  Sn_2$>  and  so  on — which 
figure  in  the  interpretations  by  thought.  We  are  going  to 
see  that  this  enterprise  of  imagination  succeeds  fairly 
well  if  n  be  small,  that  its  prosperity  decreases  as  n 
increases,  and  that  its  failure  is  well-nigh  complete  when 
n  is  taken  very  large.  At  the  same  time,  we  shall  see 
that,  in  the  case  of  interpretations  by  thought,  symmetry 
never  fails  in  even  the  least  degree,  no  matter  how  high 
the  dimensionality  of  the  space  in  which  we  are  operating. 

Let  us  for  convenience  denote  any  two  reciprocal 
thought-interpretations  by  the  symbols  T(u)  and  T(x)> 
the  former  when  the  -m's  are  fixed  and  the  x's  are  variable, 
and  the  latter  when  the  x's  are  fixed  and  the  us  are 
variable;  and  let  I(u)  and  I(x)  denote  the  corresponding 
pair  of  interpretations  essayed  by  imagination. 

Consider  first  the  simple  case  where  w=2;  S2,  the 
space  of  operation,  is  a  plane;  equation  (1)  now  is: 
(1)  U1X1+U2X2  +  1  =0.  What  are  T{u)  and  T(x)1  The 
former  is  a  conceptual  range  of  points;  the  latter,  a 
conceptual    pencil   of  lines.     What    are    I(u)    and    I(x)  ? 


THE    PSYCHOLOGY    OF    MATHEMATICS      419 

The  former  is  the  conceptual  range's  so-called  image  or 
mental  picture,  commonly  represented  to  the  physical 
eye  by  a  row  or  series  of  dots;  the  latter  is  the  conceptual 
pencil's  so-called  image  or  mental  picture,  commonly 
represented  to  the  physical  eye  by  a  set  of  physical  lines 
(or  dot  rows)  having  a  physical  point  (or  dot)  in  common. 
You  are  familiar  with  the  sensuous  figures. 

Still  keeping  n  =  2,  let  us  take  a  pair  of  equations  like 
(i),  writing  them,  for  short,  (2)  E'(u,  x)  =  0,  (3) 
E"(u,  *)=0;  and  consider  (4)  E'  +  \E"  =  0  where  X 
is  a  parameter.  Denote  the  ranges  represented  by  (2)  and 
(3)  by  R'  and  it",  and  denote  the  pencils  represented  by 
them  by  P'  and  P" .  What  are  T{u)  and  T(x)  of  (4)  ? 
The  former  is  a  conceptual  variable  range  of  the  pencil 
(of  ranges)  determined  by  R'  and  R"\  the  latter  is  a 
conceptual  variable  point  (or  pencil)  determined  by  P'  and 
P" \  it  is  plain  that  I{u)  and  I{x)  are  respectively  the 
so-called  images  of  the  variable  range  and  variable  point 
(or  pencil)  just  mentioned. 

Advancing  to  the  case  where  n  =3,  we  have  for  field  of 
operation  the  space  S3  of  ordinary  geometry.     Consider 

(5)  U1X1+U2X2+U3X3  +  1  =  0.  T{u)  of  (5)  is  obviously  a 
conceptual  plane  of  points,  while  T{x)  is  a  conceptual 
bundle  of  planes  (a  point,  that  is,  enveloped  by  the  planes 
containing  it);  and,  of  course,  /(«)  and  I(x)  are  respect- 
ively the  "  images  "  of  the  conceptual  plane  and  bundle 
(or  point  as  the  bundle's  vertex  or  carrier). 

Let  us  now  take  a  pair  of  equations  like  (5),  namely, 

(6)  E'=0,  (7)  E"=Q;  and  consider  (8)  E'  +  \E"=0. 
Let  r'  and  tt"  be  the  planes,  and  B'  and  B"  the  bundles 
(or  points),  represented  by  (6)  and  (7).  T(u)  of  (8)  is  a 
conceptual  variable  plane  of  the  axial  pencil  (of  planes) 
determined  by  tt'  and  tt";    T(x)  is  a  conceptual  variable 


420  MATHEMATICAL   PHILOSOPHY 

point  (bundle  veitex)  of  the  line  (axis  of  plane-pencil) 
determined  by  B'  and  B";  while  Iiu)  and  I(x)  are  imag- 
ination's correspondents  of  the  foregoing  concepts. 

Finally,  let  us  join  with  (6)  and  (7)  a  third  equation 
(9)  E'"  =  0,  independent  of  them  and  representing  a 
plane  71-'"  or  a  point  (plane  bundle)  B'" .  Consider  the 
equation  (10)  E'-\-\E" +nE'"  =  0.  What  are  its  T(u)  and 
T(x)  ?  The  former  is  a  conceptual  variable  plane  of  the 
point  (or  bundle)  determined  by  vr',  ir"  and  •/";  the  latter 
is  a  conceptual  variable  point  (bundle  vertex)  of  the 
plane  determined  by  B',  B"  and  B'"\  while  I(u)  and 
I{x)  are  the  imitating  "  images  "  of  the  same. 

Let  us  now  pass  to  n—\\  our  field  of  operation  is 
S4,  a  four  -  dimensional  space  of  points.  Consider 
(11)  u\X\-\-u<>x<2.-\-u?,xz-\-u±x±-\-\  =0.  Its  T(u)  is  a  con- 
ceptual lineoid  (an  S3)  of  points,  and  its  T(x)  is  a  concept- 
ual hypersheaf  of  lineoids  (a  point  enveloped  by  the 
00 3  lineoids  containing  it).  Now  scrutinize  carefully  the 
results,  I(u)  and  I(x),  of  imagination's  effort  to  imitate 
or  represent  pictorially  the  concepts  T(u)  and  T{x). 
You  observe  at  once  the  following  facts:  (a)  both  I(u) 
and  I(x)  are  inferior  to  their  analogues  for  «=3  or  2; 
(b)  the  defect  of  I(u)  differs  in  kind  from  that  of  I(x). 
Indeed  the  two  kinds  of  defect  are,  in  a  sense,  reciprocal; 
for  I(u),  in  trying  to  match  T(u),  though  it  succeeds  in 
imaging  points  and  point  configurations  interior  to  the 
lineoid  or  locus,  presents  no  image  of  the  lineoid  itself 
or  the  locus  as  a  whole;  while,  on  the  other  hand,  I(x)y 
in  trying  to  match  T(x)>  presents  an  image  corresponding 
to  the  point  or  envelope  but  no  image  to  match  the 
enveloping  lineoids.  The  contrast  may  be  vividly  seen 
as  follows:  Note  that,  in  the  one  case,  the  lineoid  is  the 
bond  or  lien  of  the  elements, — points, — of  which  it  is  the 
locus,  and  that,  in  the  other  case,  the  point  is  the  bond  or 


THE    PSYCHOLOGY    OF    MATHEMATICS      421 

lien  of  the  elements, — lineoids, — of  which  it  is  the  envelope. 
And  now  the  fact  to  be  noticed  is  this:  I{u)  images  ele- 
ments, but  not  their  bond;  I(x)  images  the  bond,  but 
not  the  elements. 

It  is  plain,  too,  that  I(u)  is  more  satisfactory  than 
I(x).  This  fact  becomes  obtrusively  evident  if  we 
geometrize  T(u)  and  T(x)  themselves.  The  two  geome- 
tries,— which  we  must  remember  are  conceptual,— match 
each  other  in  fact-to-fact  fashion  perfectly;  with  respect 
to  each  other  they  are  perfectly  symmetric.  In  the  two 
geometries  a  point  of  T{u)  corresponds  to  a  lineoid  of 
T{x)y  and  a  line-segment  joining  two  points  of  T(u) 
corresponds  to  the  angle  of  two  lineoids  of  T(x).  Now 
it  is  evident  that  the  image  of  a  segment  is  very  superior 
to  any  image  we  can  form  for  the  angle  between  two 
intersecting  lineoids.  I  need  not  give  further  examples, 
which  are  endless  in  number  and  tell  the  same  tale.  If 
you  desire  to  do  so,  you  can  pursue  the  matter  in  S4 
and  in  higher  and  higher  spaces. 

The  conclusion  is  that,  in  relation  to  space,  conception 
or  thought  is  perfectly  symmetric  and  that  imagination  or 
intuition  is  asymmetric.  As  n  increases,  thought  con- 
tinues to  look  about  in  spaces  of  ever-ascending  dimen- 
sionality like  a  binocular  being  with  no  impairment  of  its 
twofold  vision;  its  light  is  spread  abroad  equally  every- 
where; whilst  imagination's  eyes  not  only  fail  more  and 
more  as  n  mounts  higher,  but  they  fail  in  unequal  measure. 
To  change  the  figure,  thought  enters  and  moves  about 
freely  in  the  hyperspaces  like  an  eagle  with  both  wings 
equally  outspread  and  always  adequate  for  any  zone 
however  vast  or  high,  but  the  movement  of  imagination 
there  is  like  the  flight  of  a  bird  of  feeble  and  failing  wings, 
unable  to  rise  and  soar. 


LECTURE  XX 
Korzybski's  Concept  of  Man  ! 

WHAT  TIME-BINDING  MEANS DIMENSIONALITY  AND  THE 

MATHEMATICAL    THEORY    OF    LOGICAL    TYPES THE 

NATURAL  LAW  OF  CIVILIZATION  AS  AN  INCREASING 

EXPONENTIAL  FUNCTION  OF  TIME HUMAN  ETHICS 

AS  TIME-BINDING  ETHICS,   NOT  THE  SPACE-BINDING 
ETHICS  OF  ANIMALS. 

A  few  years  ago  our  lives  were  lapt  round  with  a 
civilization  so  rich  and  comfortable  in  manifold  ways, 
so  omnipresent,  so  interwoven  with  our  whole  environ- 
ment, that  we  did  not  reflect  upon  it  but  habitually  took 
it  all  for  granted  as  we  take  for  granted  the  great  gifts 
of  Nature, — land  and  sea,  light  and  sky  and  the  common 
air.  We  were  hardly  aware  of  the  fact  that  Civilization 
is  literally  a  product  of  human  labor  and  time;  we  had 
not  thought  deeply  upon  the  principle  of  its  genesis  nor 
seriously  sought  to  discover  the  laws  of  its  growth;  we 
had  not  been  schooled  to  reflect  that  we  who  were  en- 
joying it  had  neither  produced  it  nor  earned  its  goods; 
we  had  not  been  educated  to  perceive  that  we  have  it 
almost  solely  as  a  bounty  from  the  time  and  toil  of 
by-gone  generations;  we  had  not  been  disciplined  to  feel 
the  mighty  obligation  which  the  great  inheritance  imposes 

1  Part  of  this  lecture  is  found  in  my  Phi  Beta  Kappa  address  on  The 
Nature  of  Man  (Science,  Sept.  9,  1921)  and  some  of  it  in  an  article  by 
me  in  The  Pacific  Review,  Dec,  1921. 

422 


KORZYBSKI'S  CONCEPT  OF  MAN  423 

upon  us  as  at  once  the  posterity  of  the  dead  and  the 
ancestry  of  the  yet  unborn.  We  had  been  born  in  the 
midst  of  a  great  civilization,  and,  in  accord  with  our 
breeding,  we  lived  in  it  and  upon  it  like  butterflies  in  a 
garden  of  flowers,  not  to  say  as  "maggots  in  a  cheese." 
Since  then  a  change  has  come.  The  World  War 
awoke  us.  The  awakening  was  rude  but  it  was  effectual. 
Everywhere  men  and  women  arc  now  thinking  as  never 
before,  and  they  are  thinking  about  realities  for  they 
know  that  there  is  no  other  way  to  cope  with  the  great 
problems  of  a  troubled  world.  They  have  learned,  too, 
that,  of  all  the  realities  with  which  we  humans  have  to 
deal,  the  supreme  reality  is  Man;  and  so  the  questions 
that  men  and  women  are  everywhere  asking  are  questions 
regarding  Man,  for  they  are  questions  of  ethics,  of  social 
institutions,  of  education,  of  economics,  of  philosophy,  of 
industrial  methods,  of  politics  and  government.  The 
questions  have  led  to  some  curious  results, — to  doctrines 
that  alarm,  to  proposals  that  startle, — and  we  are  wont 
to  call  them  radical,  revolutionary,  red.  Is  it  true  that 
our  thinking  has  been  too  radical?  How  the  question 
would  have  made  Plato  smile — Plato  who  had  seen  his 
venerated  teacher  condemned  to  death  for  radical  criti- 
cism. No,  the  trouble  is  that,  in  the  proper  sense  of  that 
much  abused  term,  our  thought  has  not  been  radical 
enough.  Our  questionings  have  been  eager  and  wide- 
ranging  but  our  thought  has  been  shallow.  It  has  been 
passionate  and  it  has  been  daring  but  it  has  not  been 
deep.  For,  if  it  had  been  deep,  we  could  not  have  failed, 
as  we  have  failed,  to  ask  ourselves  the  fundamental 
question:  What  is  that  in  virtue  of  which  human  beings 
are  human?  What  is  the  distinctive  place  of  our  human 
kind  in  the  hierarchy  of  the  world's  life?    What  is  Man? 


424  MATHEMATICAL   PHILOSOPHY 

I  have  called  the  question  "fundamental" — it  is 
fundamental — the  importance  of  a  right  answer  is  sov- 
ereign— for  it  is  obvious,  once  the  fact  is  pointed  out, 
that  the  character  of  human  historys  the  character  of 
human  conduct,  and  the  character  of  all  our  human  insti- 
tutions depend  both  upon  what  man  is  and  in  equal  or 
greater  measure  upon  what  we  humans  think  man  is. 

Why,  then,  have  we  not  asked  the  question?  The 
reason  doubtless  is  that  we  have  consciously  or  uncon- 
sciously taken  it  for  granted  that  we  knew  the  answer. 
For  why  enquire  when  we  are  sure  we  know? 

But  have  we  known?  Is  our  assumption  of  knowledge 
in  this  case  just?  Have  we  really  known,  do  we  know 
now,  what  is  in  fact  the  idiosyncrasy  of  the  human  class 
of  life?  Do  we  know  critically  what  we,  as  representa- 
tives of  man,  really  are?  Here  it  is  essential  to  dis- 
tinguish; we  are  speaking  of  knowledge;  there  is  a  kind 
of  knowledge  that  is  instinctive, — instinctive  knowledge, 
— immediate  inner  knowledge  by  instinct, — the  kind  of 
knowledge  we  mean  when  we  say  that  we  know  how  to 
move  our  arms  or  that  a  fish  knows  how  to  swim  or  that 
a  bird  knows  how  to  fly.  I  do  not  doubt  that,  in  this 
sense  of  knowing,  we  do  know  what  human  beings  are ; 
it  is  the  kind  of  knowledge  that  a  fish  has  of  what  fishes 
are  or  that  a  bird  has  of  what  birds  are.  But  there  is 
another  kind  of  knowledge, — scientific  knowledge, — 
knowledge  of  objects  by  analyzing  them, — objective 
knowledge  by  concepts, — conceptual  knowledge  of  ob- 
jects; it  is  the  kind  of  knowledge  we  mean  when  we  say 
that  we  know  or  do  not  know  what  a  plant  is  or  what  a 
number  is.  Now,  we  do  not  suppose  fish  to  have  this 
sort  of  knowledge  of  fish;  we  do  not  suppose  a  bird  can 
have    a    just   conception, — nor,    properly    speaking,    any 


KORZYBSKI'S  CONCEPT  OF  MAN  425 

conception, — of  what  a  bird  is.  We  are  speaking  of 
concepts,  and  our  question,  you  see,  is  this:  Have  we 
humans  a  just  Concept  of  Man?  If  we  have,  it  is  rea- 
sonable to  suppose  that  we  inherited  it,  for  so  important 
a  thing,  had  it  originated  in  our  time,  would  have  made 
itself  heard  of  as  a  grave  discovery.  So  I  say  that,  if 
we  have  a  just  concept  of  man,  it  must  have  come  down 
to  us  entangled  in  the  mesh  of  our  inherited  opinions 
and  must  have  been  taken  in,  as  such  opinions  are  usually 
taken  in,  from  the  common  air,  by  a  kind  of  "cerebral 
suction." 

If  we  discover  that  we  have  never  had  a  just  concept 
of  man,  the  fact  should  not  greatly  astonish  us,  for  the 
difficulty  is  unique;  man,  you  see,  is  to  be  both  the  knower 
and  the  object  known;  the  difficulty  is  that  of  a  knower 
having  to  objectify  itself  and  having  then  to  form  a  just 
concept  of  what  the  object  is. 

In  saying  that  in  the  thought  of  our  time  the  great 
question  has  not  been  asked,  I  have  now  to  make  one 
important  exception  and,  so  far  as  I  know,  only  one.2  I 
refer  to  Count  Alfred  Korzybski,  the  Polish  engineer. 
In  his  momentous  book  ( The  Manhood  of  Humanity : 
The  Science  and  Art  of  Human  Engineering*) ,  he  has 
both  propounded  the  question  and  submitted  an  answer 
that  is  worthy  of  the  serious  attention  of  every  serious 
student,  whatever  his  field  of  study.  It  is  the  aim  of  this 
lecture  to  present  the  answer  and  to  examine  it  by  help 
of  the  Theory  of  Logical  Types,  the  Theory  of  Classes, 

:  Since  writing  the  foregoing  I  have  ob-erved  a  learned  discussion  of 
the  question  by  Professor  Wm.  E.  Ritter  in  an  article,  Science  and  Or- 
ganized Civilization,  in  the  Scientific  Monthly,  Aug.,  1917.  Professor 
Ritter  once  more  defines  man  as  a  kind  of  animal  but  the  distinctive 
marks  of  the  kind,  as  given  by  him,  are  so  grave  as  to  make  one 
wonder  why  he  did  not  altogether  drop  the  "animal"  element  from  the 
definition. 

'  E.  P.  Dutton  &  Company. 


426  MATHEMATICAL    PHILOSOPHY 

and  the  author's  closely  allied  notion  of  "Dimensions." 
Let  me  say  at  the  outset  that  one  who  would  read 
the  book  understanding^  must  come  to  it  prepared  to 
grapple  with  a  central  concept,  a  concept  whose  role 
among  the  other  ideas  in  the  work  is  like  that  of  the  sun 
in  the  solar  system.  It  happens,  therefore,  that  readers 
of  the  book,  or  of  any  other  book  built  about  a  central 
concept,   fall  into  three  mutually  exclusive  classes: 

(I)  The  class  of  those  who  miss  the  central  concept 
—  (I  have  known  a  learned  historian  to  miss  it) — not 
through  any  fault  of  their  own, — they  are  often  indeed 
well  meaning  and  amiable  people, — but  simply  because 
they  are  not  qualified  for  conceptual  thinking  save  that 
of  the  commonest  type. 

(II)  The  class  of  those  who  seem  to  grasp  the 
central  concept  and  then  straightway  show  by  their 
manner  of  talk  that  they  have  not  really  grasped  it  but 
have  at  most  got  hold  of  some  of  its  words.  Intellectu- 
ally such  readers  are  like  the  familiar  type  of  undergrad- 
uate who  "flunks"  his  mathematical  examinations  but  may 
possibly  "pull  through"  in  a  second  attempt  and  so  is 
permitted,  after  further  study,  to  try  again. 

(III)  The  class  of  those  who  firmly  seize  the  central 
concept  and  who  by  meditating  upon  it  see  more  and 
more  clearly  the  tremendous  reach  of  its  implications.  If 
it  were  not  for  this  class,  there  would  be  no  science  in  the 
world  nor  genuine  philosophy.  But  the  other  two  classes 
are  not  aware  of  the  fact  for  they  are  merely  "verbalists." 
In  respect  of  such  folk,  the  "Behaviorist"  school  of 
psychology  is  right  for  in  the  psychology  of  classes  (I) 
and  (II)  there  is  no  need  for  a  chapter  on  "Thought 
Processes" — it  is  sufficient  to  have  one  on  "The  Language 
Habit." 


KORZYBSKFS  CONCEPT  OF  MAN     427 

What  is  that  central  concept?  What  is  Korzybski's 
Concept  of  Man?  I  wish  to  present  it  as  clearly  as  I 
can.  It  is  a  concept  defining  man  in  terms  of  Time. 
"Humanity,"  says  the  author,  "is  the  time-binding  class 
of  life."  What  do  the  words  mean?  What  is  meant  by 
time-binding  or  the  binding  of  time?  The  meaning, 
which  is  indeed  momentous,  will  be  clearer  to  us  if  we 
prepare  for  it  by  a  little  preliminary  reflection. 

Long  ages  ago  there  appeared  upon  this  planet — no 
matter  how — the  first  specimens  of  our  human  kind. 
What  was  their  condition?  It  requires  some  meditation 
and  some  exercise  of  imagination  to  realize  keenly  what 
it  must  have  been.  Of  knowledge,  in  the  sense  in  which 
we  humans  now  use  the  term,  they  had  none — no  science, 
no  philosophy,  no  art,  no  religion;  they  did  not  know 
what  they  were  nor  where  they  were ;  they  knew  nothing 
of  the  past,  for  they  had  no  history,  not  even  tradition; 
they  could  not  foretell  the  future,  for  they  had  no  knowl- 
edge of  natural  law;  they  had  no  capital, — no  material 
or  spiritual  wealth, — no  inheritance,  that  is,  from  the 
time  and  toil  of  by-gone  generations;  they  were  without 
tools,  without  precedents,  without  guiding  maxims,  with- 
out speech,  without  any  light  of  human  experience;  their 
ignorance,  as  we  understand  the  term,  was  almost  abso- 
lute. And  yet,  compared  with  the  beasts,  they  were 
miracles  of  genius,  for  they  contrived  to  do  the  most 
wonderful  of  all  things  that  have  happened  on  our  globe 
— they  initiated,  I  mean,  the  creative  movement  which 
their  remote  descendants  call  Civilization. 

Why?  What  is  the  secret?  Have  you  ever  tried  to 
find  it?  The  secret  is  that  those  rude  animal-resembling, 
animal-hunting,  animal-hunted  ancestors  of  ours  were  a 
new  kind  of  creature  in  the  world — a  new  kind  because 


428  MATHEMATICAL   PHILOSOPHY 

endowed  with  a  strange  new  gift — a  strange  new  capacity 
or  power — a  strange  new  energy,  let  us  call  it.  And  it 
is  in  the  world  today.  What  is  it?  We  know  it  partly 
by  its  effects  and  partly  by  its  stirring  within  us  for  as 
human  beings,  as  representatives  of  Man,  we  all  of  us 
have  it  in  some  measure.  It  is  the  energy  that  invents — 
that  produces  instruments,  ideas,  institutions  and  doc- 
trines; it  is,  moreover,  the  energy  that,  having  invented, 
criticizes,  then  invents  again  and  better,  thus  advancing 
in  excellence  from  creation  to  creation  endlessly.  Be  good 
enough  to  reflect  and  to  reflect  again  upon  the  significance 
of  those  simple  words:  invents;  having  invented,  criti- 
cizes; invents  again  and  better;  thus  advancing,  by  cre- 
ative activity,  from  stage  to  stage  of  excellence  without 
end.  Their  sound  is  familiar;  but  what  of  their  ultimate 
sense?  We  ought  indeed  to  pause  here,  withdraw  to  the 
solitude  of  some  cloister  and  there  in  the  silence  meditate 
upon  their  meaning;  for  they  do  not  describe  the  life  of 
beasts;  they  characterize  Man. 

We  are  speaking  of  a  peculiar  kind  of  energy — the 
energy  that  civilizes — that  strange  familiar  energy  that 
makes  possible  and  makes  actual  the  great  creative  move- 
ment which  we  call  human  Progress,  of  which  we  talk 
much  and  think  but  little.  Let  us  scrutinize  it  more 
closely;  let  us,  if  we  can,  lay  bare  its  characteristic  rela- 
tion to  Time  for  its  relation  to  Time  is  the  relation  of 
Time  to  the  distinctive  life  of  Man. 

Compare  some  representative  of  the  animal  world, 
a  bee,  let  us  say,  or  a  beaver,  with  a  correspondingly 
representative  man.  Consider  their  achievements  and 
the  ways  thereof.  The  beaver  makes  a  dam;  the  man, 
a  bridge  or  some  discovery, — analytical  geometry,  for 
example,  or  the  art  of  printing,  or  the  Keplerian  laws  of 


KORZYBSKI'S  CONCEPT  OF  MAN  429 

planetary  motion,  or  the  atomic  constitution  of  matter. 
The  two  achievements, — that  of  the  beaver  and  that  of 
the  man, — are  each  of  them  a  product  of  three  factors: 
time,  toil,  and  raw  material,  where  the  last  signifies,  in 
the  case  of  purely  scientific  achievement,  the  data  of 
sense,  in  which  science  has  its  roots.  Both  achievements 
endure,  it  may  be  for  a  short  while  only, — as  in  the  case 
of  the  dam  or  the  bridge, — or  one  of  them  may  endure 
endlessly, — as  in  that  of  a  scientific  discovery.  What 
happens  in  the  next  generation?  The  new  beaver  begins 
where  its  predecessor  began  and  ends  where  it  ended — 
it  makes  a  dam  but  the  dam  is  like  the  old  one.  Yet  the 
old  dam  is  there  for  the  new  beaver  to  behold,  to  con- 
template, and  to  improve  upon.  But  the  presence  of  the 
old  dam  wakes  in  the  beaver's  "mind"  no  inventive  im- 
pulse, no  creative  stirring,  and  so  there  is  no  improve- 
ment, no  progress.  Why  not?  The  answer  is  obvious: 
the  beaver  "mind"  is  such  that  its  power  to  achieve  is 
not  reinforced  by  the  presence  of  past  achievement.  The 
new  beaver's  time  is  indeed  overlapped,  in  part  or  wholly, 
by  the  time  of  its  predecessor  for  the  latter  time  is  pres- 
ent as  an  essential  factor  of  the  old  dam,  but  that  old- 
time  factor,  though  present,  produces  nothing — it  is  as 
dead  capital,  bearing  no  interest.  Such  is  the  relation 
of  the  beaver  "mind," — of  the  animal  mind, — to  time. 
Now,  what  of  the  new  man?  What  does  he  do? 
What  he  does  depends,  of  course,  upon  his  predecessor's 
achievement;  if  this  was  a  bridge,  he  makes  a  better 
bridge  or  invents  a  ship;  if  it  was  the  discovery  of  analyt- 
ical geometry,  he  enlarges  its  scope  or  invents  the  cal- 
culus; if  it  was  the  art  of  printing,  he  invents  a  printing 
press;  if  it  was  the  discovery  of  the  laws  of  planetary 
motion,  he  finds  the  law  of  gravitation;  if  it  was  the  dis- 


430  MATHEMATICAL   PHILOSOPHY 

covery  of  the  atomic  constitution  of  matter,  he  discovers 
the  electronic  constitution  of  atoms.  Such  is  the  familiar 
record — improvement  of  old  things,  invention  of  new 
ones — Progress.  Why?  Again  the  answer  is  obvious: 
the  mind  of  man,  unlike  animal  "mind,"  is  such  that  its 
power  to  achieve  is  reinforced  by  past  achievement.  As 
in  the  case  of  the  beaver,  so  in  that  of  man,  the  successor's 
time  is  overlapped  by  the  predecessor's  time  for  the  latter 
time  continues  its  presence  as  an  essential  factor  in  the 
old  achievement,  which  endures;  but, — and  this  the  point, 
— in  man's  case,  unlike  the  beaver's,  the  old-time  factor 
is  not  merely  present,  it  works;  it  is  not  as  dead  capital, 
bearing  no  interest,  and  ultimately  perishing — it  is  living 
capital  bearing  interest  not  only  but  interest  perpetually 
compounded  at  an  ever-increasing  rate.  And  the  interest 
is  growing  wealth, — material  and  spiritual  wealth, — not 
merely  physical  conveniences  but  instruments  of  power, 
understanding,  intelligence,  knowledge  and  skill,  beauti- 
ful arts,  science,  philosophy,  wisdom,  freedom — in  a 
word,  Civilization. 

That  great  process, — involving  some  subtle  alchemy 
that  we  do  not  understand, — by  which  the  /iw^-factor, 
embodied  in  things  accomplished,  perpetually  reinforces 
more  and  more  the  achieving  potency  of  the  human  mind, 
—  the  process  by  which  mysterious  Time  thus  continually 
and  increasingly  augments  the  civilizing  energy  of  the 
world, — the  process  by  which  the  evolution  of  civilization 
involves  the  storing  up  or  involution  of  time, — it  is  that 
mighty  process  which  Korzybski  happily  designates  by 
the  term,  Time-binding.  The  term  will  recur  frequently 
in  our  discussion,  and  so  I  recommend  that  you  dwell 
upon  its  meaning  as  given  until  you  have  seized  it  firmly. 
It  is  because  time-binding  power  is  not  only  peculiar  to 


KORZYBSKI'S  CONCEPT  OF  MAN  431 

man  but  is,  among  man's  distinctive  marks,  beyond  all 
comparison  the  most  significant  one — it  is  because  of  that 
two-fold  consideration  that  the  author  defines  humanity 
to  be  "the  time-binding  class  of  life." 

Such,  then,  is  Korzybski's  answer  to  the  most  im- 
portant of  all  questions:  what  is  Man?  Do  not  lose  sight 
of  the  fact  that  we  have  here  a  concept  and  that  it  defines 
man  in  terms  of  a  certain  relation,  subtle  indeed  but  un- 
doubtedly characteristic,  that  man  has  to  time.  By  saying 
that  the  relation  is  "characteristic"  of  man  I  mean  that, 
among  known  classes  of  life,  man  and  only  man  has  it. 
Animals  have  it  not  or,  if  they  have  it,  if  they  have  time- 
binding  capacity,  they  have  it  in  a  degree  so  small  that 
it  may  be  neglected  as  mathematicians  neglect  infini- 
tesimals of  higher  order. 

The  answer  in  question  is  not  one  to  which  the  world 
has  been  or  is  now  accustomed.  If  you  apply  for  an 
answer  to  the  thought  of  the  bygone  centuries  or  to  the 
regnant  philosophies  of  our  own  time,  what  answer  will 
you  get?  It  will  be  one  or  the  other  of  two  kinds:  it 
will  be  a  zoological  answer — man  is  an  animal  a  kind 
or  species  of  animal,  the  bete  humaine;  or  it  will  be  a 
mythological  answer — man  is  a  mysterious  compound  or 
union  of  animal  (a  natural  thing)  with  something  "super- 
natural." Such  are  the  rival  conceptions  now  current 
throughout  the  world.  They  have  come  to  us  as  a  part 
of  our  philosophical  inheritance.  Some  of  us  hold  one 
of  them;  some  of  us,  the  other;  and  no  doubt  many  of 
us  hold  both  of  them  for,  though  they  are  mutually  in- 
compatible, the  mere  incompatibility  of  two  ideas  does 
not  necessarily  prevent  them  from  finding  firm  lodgment 
in  a  same  brain. 

That  Korzybski's  concept  of  man  is  just  and  impor- 


432  MATHEMATICAL   PHILOSOPHY 

tant, — entirely  just  and  immeasurably  important, — I  have 
no  reason  to  doubt  after  having  meditated  much  upon  it. 
But  the  author  does  not  content  himself  with  presenting 
that  concept;  he  goes  much  further;  he  denies  outright 
the  zoological  conception  and  similarly  denies  the  ages- 
old  rival,  the  mythological  conception,  denouncing  both 
of  them  as  being  at  once  false  to  fact  and  vicious  in  effect. 

Why  false?    Wherein? 

Let  us  deal  first  with  the  zoological  or  biological  con- 
ception. Natural  phenomena  are  to  be  conceived  and 
defined  in  accord  with  facts  revealed  by  observation  and 
analysis.  The  phenomena  the  author  is  concerned  with 
are  the  great  life-classes  of  the  world:  plants,  animals, 
and  humans.  What,  he  asks,  are  the  significant  facts 
about  them,  their  patent  cardinal  relations,  their  dis- 
tinctive marks,  positive  and  negative?  And  his  answer 
runs  as  follows:  Of  plants  the  most  significant  positive 
mark  is  their  power  to  "bind"  the  basic  energies  of  the 
world — to  take  in,  transform  and  appropriate  the  ener- 
gies of  sun,  soil,  water  and  air;  but  they  lack  autonomous 
power  to  move  about  in  space,  and  that  lack  is  a  highly 
significant  negative  mark  of  plants.  The j^hmts^are  said 
to  constitute  the  "chemistry-binding"  or  basic-energy- 
binding  class  of  life;  the  name  suggests  only  the  positive 
mark  but  it  is  essential  to  note  that  the  definition  of  the 
class  is  effected  by  the  positive  and  the  negative  marks 
conjoined.  What  of  the  animals?  These,  like  the  plants, 
take  in,  transform  and  appropriate  the  basic  energies  of 
sun,  soil,  water  and  air,  taking  them  in  large  part  as 
already  transformed  by  the  plants;  but  this  power  of 
animals  to  bind  basic  energies, — the  positive  one  of  the 
two  defining  marks  of  plants, — is  not  a  defining  mark 
of  animals;  the  positive  defining  mark  of  animals  is  their 


KORZYBSKFS  CONCEPT  OF  MAN     433 

autonomous  power  to  move  x  about  in  space, — to  crawl  or 
run  or  fly  or  swim, — enabling  them  to  abandon  one  place 
and  occupy  another  and  so  to  harvest  the  natural  fruits 
of  many  localities;  this  positive  mark,  you  observe,  is  a 
relation  of  animals  to  space;  but  they  have,  we  have  seen, 
a  negative  mark,  a  relation  to  time — animals  lack  capacity 
for  binding  time.  Because  of  the  positive  mark,  animals 
are  said  to  constitute  the  "space-binding"  class  of  life, 
but  it  is  to  be  carefully  noted  that  the  definition  (as  dis- 
tinguished from  the  name)  of  the  class  is  effected  by 
the  positive  mark  conjoined  with  the  negative  one. 
Finally,  what  of  humans?  We  have  already  seen  the 
answer  and  the  ground  thereof — humanity  is  the  time- 
binding  class  of  life.  For  the  sake  of  clarity  let  us 
summarize  the  conceptions,  or  definitions,  as  follows:  a 
plant  is  a  living  creature  having  the  capacity  to  bind 
basic  energies  and  lacking  the  autonomous  ability  to 
move  in  space;  an  animal  is  a  living  creature  having  the 
autonomous  ability  to  move  about  in  space  and  lacking 
the  capacity  for  binding  time;  a  man,  or  a  human,  is  a 
living  creature  having  time-binding  power. 

It  is  to  be  noted  that,  as  thus  conceived,  the  great 
life-classes  of  the  world  constitute  a  hierarchy  arranged 
according  to  a  principle  which  Korzybski  calls  life-dimen- 
sions or  dimensionality,  as  follows: 

The  plants,  or  basic-energy-binders,  belong  to  the 
lowest  level  or  type  of  life  and  constitute  the  life-dimen- 
sion /. 

The  animals,  or  space-binders,  belong  to  the  next 
higher  level  or  type  of  life  and  constitute  the  life-dimen- 
sion //. 

"Do  sessile  animals  really  constitute  an  exception?     It  can  lie  shown, 
I  think,  that  such  animals  are  space-binders  in   Korzybski's  sense. 


434  MATHEMATICAL   PHILOSOPHY 

Human  beings,  or  time-binders,  belong  to  a  still 
higher  level  or  type  of  life  and  constitute  the  life-dimen- 
sion ///. 

Whether  there  be  a  yet  higher  class  of  life  we  do 
not  know  and  that  is  why  in  the  conception  of  man  no 
negative  mark  is  present. 

Now,  it  is,  of  course,  perfectly  clear  that,  accord- 
ing to  the  foregoing  conceptions  or  definitions,  the  old 
zoological  conception  of  man  as  a  species  of  animal  is 
false,  as  the  author  contends.  But  may  we  not  say  that 
he  is  here  merely  playing  with  words?  Is  it  not  entirely 
a  matter  of  arbitrary  definition?  Has  he  not,  merely 
to  please  his  fancy,  quite  willfully  defined  the  term 
"animal"  in  such  a  way  as  to  exclude  humans  from  the 
class  so  defined?  The  answer  is  undoubtedly,  No.  Of 
course,  it  goes  without  saying  that  we  could,  if  we  chose, 
define  the  mere  word  "animal"  or  any  other  noun  so  as 
to  make  it  stand  for  the  "class"  of  plants,  elephants, 
humans,  jabberwocks  and  newspapers.  But  we  do  not  so 
choose.  Why  not?  Because  we  desire  our  definitions 
to  be  expedient,  to  be  helpful,  to  serve  the  purpose  of 
rational  thinking.  We  want  them,  in  other  words,  to 
correspond  to  facts.  Let  us,  then,  forget  the  word  for 
a  little  while  and  look  at  the  facts.  It  is  a  fact  that  there 
is  a  class  of  creatures  having  space-binding  capacity  but 
not  time-binding  capacity;  it  is  a  fact  that  there  is  another 
class  of  creatures  having  both  kinds  of  capacity;  it  is  a 
fact  that  the  difference  between  the  two, — namely,  the 
capacity  for  binding  time, — is  not  only  beyond  all  com- 
parison the  most  significant  of  the  marks  peculiar  to  man, 
but  is  indeed  the  most  significant  and  precious  thing  in 
the  world;  it  is,  therefore,  a  fact  that  not  only  the  inter- 
ests of  sound  ethics,  but  the  interests  of  science,  demand 


KORZYBSKI'S    CONCEPT   OF   MAN  433 

that  the  two  classes,  thus  distinct  by  an  infinite  difference 
of  kind  of  endowment,  be  not  intermixed  in  thought  and 
discourse;  it  is  a  fact  that  use  of  the  same  term  "animal" 
to  denote  the  members  of  both  classes, — men  and  beasts 
alike, — constantly,  subtly,  powerfully  tends  to  produce 
both  intellectual  and  moral  obfuscation;  it  is,  therefore, 
a  fact  that  the  author's  condemnation  of  the  zoological 
conception  as  false  to  fact  is  amply  justified  on  the  best 
of  grounds. 

It  is  indeed  true  that  humans  have  certain  animal 
organs,  animal  functions,  and  animal  propensities,  but  to 
say  that,  therefore,  humans  are  animals  is  precisely  the 
same  kind  of  logical  blunder  as  we  should  commit  if  we 
said  that  animals  or  humans  are  plants  because  they  have 
certain  organs,  functions  and  properties  in  common  with 
plants;  and  the  blunder  is  of  a  kind  that  is  fundamental 
— it  is  the  kind  which  mathematicians  call  the  confusion 
of  types  or  of  classes  and  which  Korzybski  calls  the 
"mixing  of  dimensions."  To  say  that  humans  are  animals 
because  they  have  certain  animal  propensities  is  logically 
on  a  par  with  saying  that  geometric  solids  are  surfaces 
because  they  have  certain  surface  properties  or  with  say- 
ing that  fractions  are  whole  numbers  because  they  have 
certain  properties  that  whole  numbers  have. 

Why  is  it  that  people  are  shocked  on  encountering  for 
the  first  time  a  categorical  denial  of  their  belief  that  man 
is  a  species  of  animal?  Do  they  feel  that  their  proper 
dignity  as  human  beings  is  thus  assailed?  Is  it  because 
the  animal  basis  of  their  space-binding  ethics  is  being  thus 
attacked?  Is  it  that  a  well-reasoned  scientific  conviction 
is  suddenly  contradicted?  I  do  not  think  the  shock  is 
due  to  any  of  these  things.  It  is,  I  believe,  due  simply 
to  the  fact  that  an  old  unquestioned,  uncrlticized  creed 


436  MATHEMATICAL    PHILOSOPHY 

of  that  great  dullard, — Common  Sense, — has  been  un- 
expectedly challenged.  For  it  is  evident  to  common 
sense, — it  is  obtrusively  evident  to  sense-perception, — 
that  humans  have  certain  animal  organs  and  animal  ex- 
perience— they  are  begotten  and  born,  they  feed  and 
grow,  have  legs  and  hair,  and  die,  all  just  like  animals; 
on  the  other  hand,  their  time-binding  faculty  is  not  thus 
evident;  it  is  not,  I  mean,  a  tangible  organ;  it  is  an  in- 
tangible function,  subtle  as  spirit;  and  so  common  sense, 
guided  according  to  its  wont  by  the  uncriticized  evidence 
of  sense,  and  thoughtlessly  taking  for  major  premise  the 
false  proposition  that  whatever  has  animal  organs  and 
propensities  is  an  animal,  concludes  that  our  human  kind 
is  a  kind  of  animal.  But  in  this  matter,  as  in  so  many 
others,  the  old  dullard  is  wrong.  The  proper  life  of 
animals  is  life-in-space;  the  distinctive  life  of  humans  is 
life-in-time. 

But  why  are  mere  concepts  so  important?  Our  lives, 
we  are  told,  are  not  controlled  by  concepts  but  by  im- 
pulses, instincts,  desires,  passions,  appetites.  The  answer 
is:  Because  concepts  are  never  "mere"  concepts  but  are, 
in  humans,  vitally  connected  with  impulses,  instincts, 
desires,  passions,  and  appetites;  concepts  are  the  means 
by  which  Reason  does  its  work,  leading  to  prosperity  or 
disaster  according  as  the  concepts  be  true  or  false. 

I  have  said  that  the  ancient  and  modern  rival  of  the 
zoological  conception  of  man  is  the  mythological  concep- 
tion according  to  which  man  is  a  mysterious  compound 
or  hybrid  of  natural  (animal)  and  supernatural.  This 
conception  might  well  be  treated  today  as  it  was  treated 
yesterday  by  Plato  (in  the  Timaeus,  for  example).  "We 
must  accept,"  said  he,  "the  traditions  of  the  men  of  old 
time  who  affirm  themselves  to  be  the   offspring  of  the 


KORZYBSKI'S  CONCEPT  OF  MAN  437 

gods — that  is  what  they  say — and  they  must  surely  have 
known  their  own  ancestors.  How  can  we  doubt  the  word 
of  the  children  of  the  gods?  Although  they  give  oo 
probable  or  certain  proofs,  still,  as  they  declare  that  they 
are  speaking  of  what  took  place  in  their  own  family,  we 
must  conform  to  custom  and  believe  them."  *  But  this 
gentle  irony, — the  way  of  the  Greek  philosopher, — is  not 
the  way  of  the  Polish  engineer.  The  latter  is  not  indeed 
without  a  blithesome  sense  of  humor  but  in  this  matter 
he  is  tremendously  in  earnest,  and  he  bluntly  affirms, 
boldly  and  confidently,  that  the  mythological  conception 
of  man  is  both  false  and  vicious.  As  to  its  validity  or 
invalidity,  it  involves,  he  says,  the  same  kind  of  logical 
blunder  as  the  zoological  conception — it  involves,  that  is, 
a  fatal  confusion  of  types,  or  mixing  of  dimensions.  To 
say  that  man  is  a  being  so  inscrutably  constituted  that  he- 
must  be  regarded  as  partly  natural  (partly  animal)  and 
partly  supernatural  (partly  divine)  is  logically  like  saying 
that  a  geometrical  solid  is  a  thing  so  wonderful  that  it 
must  certainly  be  a  surface  miraculously  touched  by  some 
mysterious  influence  from  outside  the  universe  of  space. 
Among  the  life-classes  of  the  world,  our  humankind  is 
the  time-binding  class;  and  Korzybski  stresses  again  and 
again  the  importance  of  recognizing  that  time-binding 
energy  and  all  the  phenomena  thereof  are  perfectly 
natural — that  Newton,  for  example,  or  Confucius,  was 
as  thoroughly  natural  as  an  eagle  or  an  oak. 

What  does  he  mean  by  "natural"?  He  lias  not  told 
us, — at  all  events,  not  explicitly, — and  that  omission  is 
doubtless  a  defect  which  ought  to  be  remedied  in  a  future 
edition  of  the  book. 

You  are  aware  that  the  terms  "nature"  and  "natural" 

1  Jowett's  translation. 


438  MATHEMATICAL    PHILOSOPHY 

are  currently  employed  in  a  large  variety  of  senses — 
most  of  them  so  vague  as  to  be  fit  only  for  the  use  of 
"literary"  men,  not  for  the  serious  use  of  scientific  men. 
What  ought  we  to  mean  by  the  term  "natural"  in  such 
a  discussion  as  we  are  now  engaged  in?  The  question 
admits,  I  believe,  of  a  brief  answer  that  is  fairly  satis- 
factory. Everyone  knows  that  the  things  encountered 
by  a  normal  human  in  the  course  of  his  experience  differ 
widely  in  respect  of  vagueness  and  certitude;  some  of 
them  are  facts  so  regular,  so  well  ascertained,  so  indubit- 
able that  they  guide  in  all  the  affairs  of  practical  life; 
they  are  known  facts,  we  say,  and  to  disregard  them 
would  be  to  perish  like  unprotected  idiots  or  imbeciles; 
such  facts  are  of  two  kinds:  facts  of  sense-perception,  or 
of  this  and  memory,  and  facts  of  pure  thought;  the 
former  are  familiar  in  the  moving  pageant  of  the  world 
— birth,  growth,  death,  day,  night,  land,  water,  sky, 
change  of  seasons,  and  so  on;  facts  of  pure  thought  are 
not  so  obtrusively  obvious  but  there  are  such  facts;  one 
of  them  is — "If  something  S  has  the  property  P  and 
whatever  has  P  has  the  property  P\  then  S  has  P'" 
Now,  all  such  facts  are  compatible — each  of  them  fits 
in,  as  we  say,  with  all  the  others.  I  take  it  that  what  we 
ought  to  mean  by  natural  is,  therefore,  this:  Nature  (or 
the  natural)  consists  of  all  and  only  such  things  as  are 
compatible  (consistent)  with  the  best-ascertained  facts 
of  sense  and  of  thought. 

If  that  be  what  Korzybski  means  by  "natural," — and 
I  think  it  very  probably  is, — then  I  fully  agree  with  him 
that  humans  are  thoroughly  natural  beings,  that  time- 
binding  energy  is  a  natural  kind  of  energy,  and  that  his 
strenuous  objection  to  the  mythological  conception  of 
man  is,  like  his  objection  to  the  zoological  conception, 


KORZYBSKI'S  CONCEPT  OF  MAN     439 

well  taken.  If  it  were  a  question  of  biological  data,  mere 
mathematicians  would,  of  course,  like  other  sensible  folk, 
defer  to  the  opinion  of  biologists;  it  is  not,  however,  a 
question  of  biological  data,  these  are  not  in  dispute;  it  is 
a  question  of  the  logical  significance  of  such  data;  and 
respecting  a  question  of  logic,  even  biologists, — for  they, 
too,  are  sensible  folk, — will  probably  admit  that  engineers 
and  mere  mathematicians  are  entitled  to  be  heard. 

In  this  connection  I  desire  to  say  that,  for  straight 
and  significant  thinking,  the  importance  of  avoiding  what 
Korzybski  calls  "mixing  dimensions"  can  not  be  over- 
stressed.  The  meaning  of  the  term  "dimensions"  as  he 
uses  it  is  unmistakable;  he  has  not,  however,  elaborated 
an  abstract  theory  of  the  idea;  such  an  elaboration  would, 
I  believe,  show  that  the  idea  is  reducible  or  nearly  reduc- 
ible to  that  of  the  Theory  of  Logical  Types,  briefly  dealt 
with  in  a  previous  lecture  and  fully  outlined  in  the  Pr'in- 
cipia  Mathematica  of  Whitehead  and  Russell;  it  is, 
moreover,  very  closely  allied  to,  if  it  be  not  essentially 
identical  with,  Professor  J.  S.  Haldane's  doctrine  of 
"categories"  as  set  forth  in  his  very  stimulating  and  sug- 
gestive book  Mechanism,  Life,  and  Personality  (E.  P. 
Dutton  and  Co.)  wherein  the  eminent  physiologist  main- 
tains that  mechanism,  life,  and  personality  belong  to  dif- 
ferent categories  constituting  a  genuine  hierarchy  such 
that  the  higher  is  not  reducible  to  the  lower,  that  life, 
for  example,  cannot  be  understood  fully  in  terms  of 
mechanism,  nor  personality  in  terms  of  life.  It  is,  you 
observe,  an  order  of  ideas  similar  to  that  of  Korzybski's 
thesis  that  humans  can  be  no  more  explained  in  terms  of 
animals  than  animals  in  terms  of  plants  or  plants  in  terms 
of  minerals.  And  it  is  an  order  of  ideas  that  recommends 
itself,  to  me  at  all  events,  because  it  is  fortified  by  the 


440  MATHEMATICAL   PHILOSOPHY 

analogous  consideration  that  geometry  cannot  be  reduced 
to  arithmetic,  nor  dynamics  to  geometry,  nor  physics  to 
dynamics,  nor  psychology  to  physics.  It  will,  I  believe, 
be  a  great  advantage  to  science  and  to  philosophy  to 
recognize  that  there  exists,  whether  we  will  or  no,  a 
hierarchy  of  categories  and  to  recognize  that,  to  an 
understanding  of  the  higher  categories,  the  lower  ones, 
though  necessary,  are  not  sufficient. 

Is  there  not,  indeed,  a  highly  important  sense  in  which 
the  phenomena  of  a  higher  category  throw  as  much  light 
upon  those  of  a  lower  as  the  latter  throw  upon  the 
former?  Who  can  deny  that,  for  example,  dynamics 
illuminates  geometry  quite  as  much  as  geometry  illumi- 
nates dynamics? 

In  Korzybski's  indictment  of  the  zoological  and  myth- 
ological conceptions  of  man  there  are,  we  have  seen,  two 
counts:  he  denies  that  the  conceptions  are  true;  and  he  de- 
nounces them  as  vicious  in  their  effects,  contending  that 
they  are  mainly  responsible  for  the  dismal  things  of 
human  history  and  for  what  is  woeful  in  the  present  plight 
of  the  world.  Of  the  former  count  I  have  already  spoken; 
respecting  the  latter  one,  my  convictions  are  as  follows; 
( i )  if  humanity  be  not  a  thoroughly  natural  class  of  life, 
the  term  "natural"  having  the  sense  above  defined,  it  is 
perfectly  evident  that  there  never  has  been  and  never  can 
be  a  system  of  human  ethics  having  the  understandability, 
the  authority,  and  the  sanction  of  natural  law,  and  this 
means  that,  under  the  hypothesis,  there  never  has  been 
and  never  can  be  an  ethical  system  "compatible  with  the 
best-ascertained  facts  of  sense  and  of  thought";  (2)  if, 
although  our  human  kind  be  in  fact  a  thoroughly  natural 
class,  we  continue  to  think  that  such  is  not  the  case,  the 
result  will  be  much  the  same — our  ethics  will  continue  to 


KORZYBSKI'S    CONCEPT   OF    MAN  441 

carry  the  confusion  and  darkness  due  to  the  presence  in 
it  of  mythological  elements;  (3)  on  the  other  hand,  so 
long  as  we  continue  to  regard  man  as  a  species  of  animal, 
the  social  life  of  the  world  in  all  its  aspects  will  continue 
to  reflect  the  tragic  misconception,  and  our  ethics  will 
remain, — what  it  always  has  been  in  large  measure, — an 
animal  ethics,  space-binding  ethics,  an  ethics  of  might, 
of  brutal  competition,  of  violence,  combat,  and  war. 

Why  so  much  stress  upon  ethics?  Because  ethics  is 
not  a  thing  apart;  it  is  not  an  interest  that  is  merely  co- 
ordinate with  other  interests;  it  penetrates  them  all. 
Ethics  is  a  kind  of  social  ether  which,  whether  it  be  good 
or  bad,  sound  or  unsound,  true  or  false,  pervades  life, 
private  and  public,  in  all  its  dimensions  and  forms;  and 
so,  if  ethics  be  vitiated  by  fundamentally  false  concep- 
tions of  human  nature,  the  virus  is  not  localized  but 
spreads  throughout  the  body  politic,  affecting  the  charac- 
ter of  all  activities  and  institutions, — education,  science, 
art,  philosophy,  economics,  industrial  method,  politics, 
government, — the  whole  conduct  and  life  of  a  tribe  or 
a  state  or  a  nation  or  a  world.  I  hardly  need  remind 
you  that  only  yesterday  the  most  precious  institutions  of 
civilization  were  in  great  danger  of  destruction  by  a 
powerful  state  impelled,  guided  and  controlled  by  ani- 
malistic ethics,  the  space-binding  ethics  of  beasts.  This 
is  indeed  an  unforgettable  illustration  of  the  mighty  fact, 
before  pointed  out,  that  the  character  of  human  history, 
human  conduct  and  human  institutions  depends,  not 
merely  upon  what  man  distinctively  is,  but  also  in  large 
measure,  even  decisively,  upon  what  we  humans  think 
man  is.  If  a  man  or  a  state  habitually  regards  humanity 
as  a  species  of  animal,  then  that  man  or  state  may  be 


442  MATHEMATICAL   PHILOSOPHY 

expected  to  act  betimes  like  a  beast  and  to  seek  justifica- 
tion in  a  zoological  philosophy  of  human  nature. 

In  view  of  such  considerations  it  is  a  great  pleasure 
to  turn  to  Korzybski's  concept  of  man,  for  it  is  not  only 
a  noble  conception,  as  none  can  fail  to  perceive,  but  it  is 
also,  as  we  have  seen,  undoubtedly  just.  Nothing  can 
be  more  important.  What  are  its  implications?  And 
what  are  its  bearings?  You  cannot  take  them  in  at  a 
glance — meditation  is  essential;  but,  if  you  will  medi- 
tate upon  the  concept,  you  will  find  that  the  body  of  its 
implications  looms  larger  and  larger  and  that  the  range 
of  its  bearings  grows  ever  clearer  and  wider.  Indeed 
we  may  say  of  it  what  Carlyle  said  of  Wilhelm  Meister: 
"It  significantly  tends  towards  infinity  in  all  directions." 
Let  us  reflect  upon  it  a  little.  We  shall  see  that  human 
history,  the  philosophy  thereof,  the  present  status  of  the 
world,  the  future  welfare  of  mankind,  are  all  of  them 
involved. 

The  central  concept  or  thesis  is  that  our  human  kind 
is  the  time-binding  class  of  life;  it  is,  in  other  words,  that 
there  is  in  our  world  a  peculiar  kind  of  energy,  time- 
binding  energy,  and  that  man  is  its  organ — its  sole  instru- 
ment or  agency.     What  are  its  implicates  and  bearings? 

One  of  them  we  have  already  noted.  It  is  that, 
though  we  humans  are  not  a  species  of  animal,  we  are 
natural  beings :  it  is  as  natural  for  humans  to  bind  time 
as  it  is  natural  for  fishes  to  swim,  for  birds  to  fly,  for 
plants  to  live  after  the  manner  of  plants.  It  is  as  natural 
for  man  to  make  things  achieved  the  means  to  greater 
achievements  as  it  is  natural  for  animals  not  to  do  so. 

That  fact  is  fundamental.  Another  one,  also  funda- 
mental, is  this :  time-binding  faculty, — the  characteristic 
of  humanity, — is  not  an  effect  of  civilization  but  is  its 


KORZYBSKI'S  CONCEPT  OF  MAN  443 

cause;  it  is  not  civilized  energy,  it  is  the  energy  that 
civilizes;  it  is  not  a  product  of  wealth,  whether  material 
or  spiritual  wealth,  but  is  the  creator  of  wealth,  both 
material  and  spiritual. 

I  come  now  to  a  most  grave  consideration.  Inasmuch 
as  time-binding  capacity  is  the  characterizing  mark, — 
the  idiosyncrasy, — of  our  human  kind,  it  follows  that 
to  study  and  understand  man  is  to  study  and  understand 
the  nature  of  man's  time-binding  energies;  the  laws  of 
human  nature  are  the  laws, — natural  laws, — of  these 
energies;  to  study  time-binding  phenomena, — the  phe- 
nomena of  civilization, — and  to  discover  their  laws  and 
teach  them  to  the  world,  is  the  supreme  obligation  of  sci- 
entific men,  for  it  is  evident  that  upon  the  natural  laws 
of  time-binding  must  be  based  the  future  science  and  art 
of  human  life  and  human  welfare. 

One  of  the  laws  we  know  now, — not  indeed  precisely, 
— but  fairly  well, — we  know  roughly,  I  mean,  its  general 
type, — and  it  merits  our  best  attention.  It  is  the  natural 
law  of  progress  in  time-binding — in  civilization-building. 
We  have  observed  that  each  generation  of  (say)  beavers 
or  bees  begins  where  the  preceding  one  began  and  ends 
where  it  ended;  that  is  a  law  for  animals,  for  mere 
space-binders — there  is  no  advancement,  no  time-binding 
— a  beaver  dam  is  a  beaver  dam — a  honey  comb  a  honey 
comb.  We  know  that,  in  sharp  contrast  therewith,  man 
invents,  discovers,  creates;  we  know  that  inventions  lead 
to  new  inventions,  discoveries  to  new  discoveries,  crea- 
tions to  new  creations;  we  know  that,  by  such  progressive 
breeding,  the  children  of  knowledge  and  art  and  wisdom 
not  only  produce  their  kind  in  larger  and  larger  families 
but  engender  new  and  higher  kinds  endlessly;  we  know 
that  this  time-binding  process,   by  which  past  time  em- 


444  MATHEMATICAL   PHILOSOPHY 

bodied  as  cofactor  of  toil  in  enduring  achievements  thus 
survives  the  dead  and  works  as  living  capital  for  aug- 
mentation and  transmission  to  posterity,  is  the  secret  and 
process  of  progressive  civilization-building.  The  ques- 
tion is:  What  is  the  Law  thereof — the  natural  law? 
What  its  general  type  is  you  apprehend  at  once ;  it  is  like 
that  of  a  rapidly  increasing  geometric  progression — if  P 
be  the  progress  made  in  a  given  generation,  conveniently 
called  the  "first,"  and  if  R  denote  the  ratio  of  improve- 
ment, then  the  progress  made  in  the  second  generation  is 
PR,  that  in  the  third  is  PR2,  and  that  made  in  the  single 
Tth  generation  will  be  PR1'  \  Observe  that  R  is  a  large 
number, — how  large  we  do  not  know, — and  that  the 
time  T  enters  as  an  exponent — and  so  the  expression 
PR7'1  is  called  an  exponential  function  of  Time,  and  it 
makes  evident,  even  to  the  physical  eye,  the  involution  of 
time  in  the  life  of  man.  This  is  an  amazing  function,  as 
every  student  of  the  Calculus  knows;  as  T  increases, 
which  it  is  always  doing,  the  function  not  only  increases 
but  it  does  so  at  a  rate  which  itself  increases  according 
to  a  similar  law,  and  the  rate  of  increase  of  the  rate  of 
increase  again  increases  in  like  manner,  and  so  on  end- 
lessly, thus  sweeping  on  towards  infinity  in  a  way  that 
baffles  all  imagination  and  all  descriptive  speech.  Yet 
such  is  approximately  the  law, — the  natural  law, — for 
the  advancement  of  Civilization,  immortal  offspring  of 
the  spiritual  marriage  of  Time  and  human  Toil.  I  have 
said  "approximately,"  for  it  does  not  represent  adequately 
the  natural  law  for  the  progress  of  civilization;  it  does 
not,  however,  err  by  excess,  it  errs  by  defect;  for,  upon 
a  little  observation  and  reflection,  it  is  evident  that  R, 
the  ratio  of  improvement,  is  not  a  constant,  as  above  con- 
templated, but  it  is  a  variable  that  grows  larger  and  larger 
as  time  increases,  so  that  the  function  PRT~X  increases 


KORZYBSKI'S  CONCEPT  OF  MAN     445 

not  only  because  the  exponent  increases  with  the  flux  of 
time,  but  because  R  itself  is  an  increasing  function  of 
time.  It  will  be  convenient,  however,  and  we  shall  not 
be  thus  erring  on  the  side  of  excess,  to  speak  of  the  above- 
mentioned  law,  though  it  is  inadequate,  as  the  natural 
law  for  the  progress  of  time-binding,  or  of  civilization- 
making. 

Hereupon,  there  supervenes  a  most  important  ques- 
tion: Has  civilization  always  advanced  in  accord  with 
the  mentioned  law?  And,  if  not,  why  not?  The  time- 
binding  energies  of  mankind  have  been  in  operation  long 
— 300,000  to  500,000  years,  according  to  the  estimates 
of  those  most  competent  to  guess — anthropologists  and 
paleontologists.  Had  progress  conformed  to  the  stated 
law  throughout  that  vast  period,  our  world  would  doubt- 
less now  own  a  civilization  so  rich  and  great  that  we 
cannot  imagine  it  today  nor  conceive  it  nor  even  con- 
jecture it  in  dreams.  What  has  been  the  trouble?  What 
have  been  the  hindering  causes?  Here,  as  you  see,  Kor- 
zybski's  concept  of  man  must  lead  to  a  new  interpretation 
of  history — to  a  new  philosophy  of  history.  A  funda- 
mental principle  of  the  new  interpretation  must  be  the 
fact  which  I  have  already  twice  stated, — namely,  that 
what  man  has  done  and  does  has  depended  and  depends 
both  upon  what  man  distinctively  is  and  also,  in  very 
great  measure,  upon  what  the  members  of  the  race  have 
thought  and  think  man  is.  We  have  here  two  determin- 
ing factors — what  man  is  and  what  we  humans  think  man 
is.  It  is  their  joint  product  which  the  sociologist  or  the 
philosophic  historian  must  examine  and  explain.  In  view 
of  the  second  factor,  which  has  hardly  ever  been  noticed 
and  has  never  been  given  its  due  weight,  Kor/.ybski,  in 
answer  to  our  question,  maintains  that  the  chief  causes 
which  have  kept  civilization   from  advancing  in   accord 


446  MATHEMATICAL    PHILOSOPHY 

with  its  natural  law  of  increase  are  man's  misconceptions 
of  man.  All  that  is  precious  in  present  civilization  has 
been  achieved,  in  spite  of  them,  by  the  first  factor — by 
what  man  is — the  peculiar  organ  of  the  civilizing  energies 
of  the  world.  It  is  the  second  factor  that  has  given 
trouble.  Throughout  the  long  period  of  our  race's  child- 
hood, from  which  we  have  not  yet  emerged,  the  time- 
binding  energies  have  been  hampered  by  the  false  belief 
that  man  is  a  species  of  animal  and  hampered  by  the 
false  belief  that  man  is  a  miraculous  mixture  of  natural 
and  supernatural.  These  are  cave-man  conceptions.  The 
glorious  achievements  of  which  they  have  deprived  the 
world  we  cannot  now  know  and  may  never  know,  but  the 
subtle  ramifications  of  their  positive  evils  can  be  tracec 
in  a  thousand  ways.  And  it  is  not  only  the  duty  of  pro- 
fessional historians  to  trace  them,  it  is  your  duty  and 
mine.  Whoever  performs  the  duty  will  be  appalled,  for 
he  will  discover  that  those  evils — the  evils  of  "magic  and 
myth,"  of  space-binding  "ethics,"  of  zoological  "right- 
eousness"— for  centuries  growing  in  volume  and  momen- 
tum— did  but. leap  to  a  culmination  in  the  World  War, 
which  is  thus  to  be  viewed  as  only  a  bloody  demonstra- 
tion of  human  ignorance  of  human  nature. 

We  are  here  engaged  in  considering  some  of  the 
major  implicates  and  bearings  of  the  new  concept  of  man. 
The  task  demands  a  large  volume  dealing  with  the  rela- 
tions of  time-binding  to  each  of  the  cardinal  concerns  of 
individual  and  social  life — ethics,  education,  economics, 
medicine,  law,  political  science,  government,  industry, 
science,  art,  philosophy,  religion.  Perhaps  you  will  write 
such  a  work  or  works.  In  the  closing  words  of  this  lec- 
ture I  can  do  no  more  than  add  to  what  I  have  said  a 
few  general  questions  and  hints. 


KORZYBSKI'S  CONCEPT  OF  MAN     447 

Korzybski  believes  that  the  great  war  marks  the  end 
of  the  long  period  of  humanity's  childhood  and  the  be- 
ginning of  humanity's  manhood.  This  second  period,  he 
believes,  is  to  be  initiated,  guided,  and  characterized  by 
a  right  understanding  of  the  distinctive  nature  of  Man. 
Is  he  over-enthusiastic?  I  do  not  know.  Time  will  tell. 
I  hope  he  is  not  mistaken.  If  he  is  not,  there  will  be 
many  changes  and  many  transfigurations. 

I  have  spoken  of  ethics  and  must  do  so  again,  for 
ethics,  good  or  bad,  is  the  most  powerful  of  influences, 
pervading,  fashioning,  coloring,  controlling  all  the  moods 
and  ways  and  institutions  of  our  human  world.  What 
is  to  be  the  ethics  of  humanity's  manhood?  It  will  not 
be  an  ethics  based  upon  the  zoological  conception  of  man; 
it  will  not,  that  is,  be  animalistic  ethics,  space-binding 
ethics,  the  ethics  of  beasts  fighting  for  "a  place  in  the 
sun,"  the  ethics  of  might,  crowding,  and  combat;  it  will 
not  be  a  "capitalistic"  ethics  lusting  to  keep  for  self,  nor 
"proletarian"  ethics  lusting  to  get  for  self;  it  will  not  be 
an  ethics  having  for  its  golden  rule  the  law  of  brutes — 
survival  of  the  fittest  in  the  sense  of  the  strongest. 
Neither  will  it  be  an  ethics  based  upon  a  mythological 
conception  of  man;  it  will  not,  that  is,  be  a  lawless  ethics 
cunningly  contrived  for  traffic  in  magic  and  myth.  It  will 
be  a  natural  ethics  because  based  upon  the  distinctive 
nature  of  mankind  as  the  time-binding, — civilization-pro- 
ducing,— class  of  life;  it  will  be,  that  is,  a  scientific  ethics 
having  the  understandability,  the  authority,  and  the  sanc- 
tion of  natural  law,  for  it  will  be  the  embodiment,  the 
living  expression,  of  the  laws, — natural  laws, — of  the 
time-binding  energies  of  man;  human  freedom  will  be 
freedom  to  live  in  accord  with  those  laws  and  righteous- 
ness will  be  the  quality  of  a  life  that  does  not  contravene 


448  MATHEMATICAL   PHILOSOPHY 

them.  The  ethics  of  humanity's  manhood  will  thus  be 
natural  ethics,  an  ethics  compatible  with  the  best-ascer- 
tained facts  of  sense  and  of  thought — it  will  be  time- 
binding  ethics — and  it  will  grow  in  solidarity,  clarity,  and 
sway  in  proportion  as  science  discovers  the  laws  of  time- 
binding, — the  laws,  that  is,  of  civilization-growth, — and 
teaches  them  to  the  world. 

And  so  I  am  brought  to  say  a  word  respecting  educa- 
tion. In  humanity's  manhood,  education, — in  home,  in 
school,  in  church, — will  have  for  its  supreme  obligation, 
and  will  keep  the  obligation,  to  teach  the  young  the  dis- 
tinctive nature  of  man  and  what  they,  as  members  and 
representatives  of  the  race  of  man,  essentially  are,  so 
that  everywhere  throughout  the  world  men  and  women 
will  habitually  understand,  because  bred  to  understand, 
what  time-binding  is,  that  their  proper  dignity  as  humans 
is  the  dignity  of  time-binding  life,  and  that  for  humans 
to  practice  space-binding  ethics  is  a  monstrous  thing,  in- 
volving the  loss  of  their  human  birthright  by  descent  to 
the  level  of  beasts.1  It  is  often  said  that  ethics  is  a  thing 
which  it  is  impossible  to  leach.  Just  the  opposite  is  true 
— it  is  impossible  not  to  teach  ethics,  for  the  teaching  of 
it  is  subtly  carried  on  in  all  our  teaching,  whether  con- 
sciously or  not,  being  essentially  involved  in  the  teacher's 
"philosophy  of  human  nature."  Every  home  or  school 
in  which  that  philosophy  is  zoological  is,  consciously  or 
unconsciously,  a  nursery  of  animalistic  ethics;  every  home 
or  school  in  which  there  prevails  a  mythological  phi- 
losophy of  human  nature  is,  consciously  or  unconsciously, 
a  nursery  of  a  lawless  ethics  of  myth  and  magic.     From 

1  In  a  recent  bulletin  of  the  Cora  L.  Williams  Institute  for  Creative 
Education,  Miss  Williams  has  said,  with  fine  insight,  that  "time-binding 
should  be  made  the  basis  of  all  instruction  and  The  Manhood  of  Hu- 
manity a  textbook  in  every  college  throughout  the  world." 


KORZYBSKI'S  CONCEPT  OF  MAN  449 

time  immemorial,  such  teaching  of  ethics,  for  the  most 
part  unconscious,  the  whole  world  has  had.  And  we  have 
seen  that  when  such  teaching  becomes  conscious,  delib- 
erate, and  organized,  a  whole  people  can  be  so  imbued 
with  both  the  space-binding  animal  ethics  of  might  and 
the  mythical  ethics  of  Gott  mit  uns  that  their  State  will 
leap  upon  its  neighbors  like  an  infuriated  beast.  Why 
should  we  not  learn  the  lesson  which  the  great  war  has 
so  painfully  taught  regarding  the  truly  gigantic  power  of 
education?  If  the  accumulated  civilization  of  many 
centuries  can  be  imperiled  by  ethical  teaching  based  upon 
a  false  philosophy  of  human  nature,  who  can  set  a  limit 
to  the  good  that  may  be  expected  from  the  conscious, 
deliberate,  organized,  unremitting  joint  effort  of  home 
and  school  and  press  to  teach  an  ethics  based  upon  the 
true  conception  of  man  as  the  agent  and  organ  of  the 
time-binding,  civilizing  energy  of  the  world?  I  cannot 
here  pursue  the  matter  further;  but  in  closing  I  should  like 
to  ask  a  few  general  questions — pretty  obvious  questions 
— indicating  roughly  the  course  which,  I  believe,  further 
enquiry  should  take. 

What  are  the  bearings  of  the  new  concept  of  man 
upon  the  social  so-called  sciences  of  economics,  politics, 
and  government? 

Can  the  new  concept  transform  those  ages-old  pseudo- 
sciences  into  genuine  sciences  qualified  to  guide  and  guard 
human  welfare  because  based  upon  scientific  understand- 
ing of  human  nature? 

In  view  of  the  radical  difference  between  the  distinct- 
ive nature  of  animals  and  the  distinctive  nature  of  man, 
what  are  likely  to  be  the  principal  differences  between 

Government  of  Space-binders,  by  Space-binders,  for 
Space-binders 


450  MATHEMATICAL   PHILOSOPHY 

and 
Government  of  Time-binders,  by  Time-binders,  for 
Time-binders? 

Which  of  the  two  kinds  of  government  best  befits  the 
social  regime  of  autocrats,  or  plutocrats,  and  slaves? 
And  which  best  befits  the  dream  of  political  equality  and 
democratic  freedom? 

Which  of  them  most  favors  the  prosperity  of  "Ac- 
quisitive Cunning"  ?  And  which  the  prosperity  of  Pro- 
ductive Skill? 

Which  of  them  is  the  most  friendly  to  the  makers  of 
wealth?    And  which  of  them  to  the  takers  thereof? 

Which  of  them  most  favors  "boss"  repression  of 
others?  And  which  makes  the  best  provision  for  intel- 
ligent self-expression? 

Which  of  them  depends  most  upon  might  and  war? 
And  which  upon  right  and  peace? 

Which  of  them  is  government  by  "politics,"  by  poli- 
ticians? And  which  of  them  by  science,  by  honest  men 
who  know? 

If  man's  time-binding  energy,  which  has  produced  all 
the  wealth  of  the  world,  both  material  and  spiritual 
wealth,  be  natural  energy,  and  if,  as  is  the  case,  the 
wealth  existing  at  a  given  moment  be  almost  wholly  a 
product  of  the  time  and  toil  of  the  by-gone  generations, 
to  whom  does  it  of  right  belong?  To  some  of  the  living? 
To  all  of  the  living?  Or  to  all  of  the  living  and  the  yet 
unborn?  Is  the  world's  heritage  of  wealth,  since  it  is  a 
natural  product  of  a  natural  energy  and  of  time  (which 
is  natural),  therefore  a  "natural  resource"  like  sunshine, 
for  example,  or  a  newfound  lake  or  land?  If  not,  why 
not?     What  is  the  difference  in  principle? 

Are  the  "right  of  conquest"  and  the  "right  of  squatter 


KORZYBSKFS  CONCEPT  OF  MAN      451 

sovereignty"  time-binding  rights?  Or  are  they  space- 
binding  "rights"  having  their  sanction  in  animalistic 
"ethics,"  in  a  zoological  philosophy  of  human  nature? 

What  are  the  bearings  of  the  new  concept  of  man 
upon  the  theory  and  practice  of  medicine?  Man,  though 
not  an  animal,  has  animal  organs  and  animal  functions. 
Are  all  the  diseases  of  human  beings  animal  diseases  or 
are  some  of  them  human  diseases,  disorders,  that  is,  af- 
fecting humans  in  their  distinctive  character  as  time- 
binders?  Can  Psycho-analysis  or  Psychiatry  throw  any 
light  upon  the  question? 

And  what  of  the  power  that  makes  for  righteousness? 
Religion,  it  would  seem,  has  the  seat  of  its  authority  in 
that  time-binding  double  relationship  in  virtue  of  which 
the  living  are  at  once  posterity  of  the  dead  and  ancestry 
of  the  unborn, — in  the  former  capacity  inheriting  as 
living  capital  the  wealth  of  civilization  from  the  time  and 
toil  of  by-gone  generations, — in  the  latter  capacity  hold- 
ing the  inheritance  in  trust  for  enlargement  and  trans- 
mission to  future  man. 

A  final  reflection:  under  the  doctrine  outlined  there 
lies  an  assumption — it  is  that,  when  men  and  women  are 
everywhere  bred  to  understand  the  distinctive  nature  of 
our  human  kind,  the  time-binding  energies  of  man  will  be 
freed  from  their  old  bondage  and  civilization  will 
advance,  in  accord  with  its  natural  law,  in  a  warless 
world,  swiftly  and  endlessly.  If  the  assumption  be  not 
true,  great  Nature  is  at  fault  and  the  world  will  continue 
to  flounder.  Of  its  truth,  there  can  be  only  one  test — 
experimentation,  trial.  The  assumption  appears  to  be 
the  only  scientific  basis  of  hope  for  the  world.  Must 
not  all  right-thinking  men  and  women  desire  ardently 
that  this  noble  assumption  be  tried? 


LECTURE    XXI 
Science  and  Engineering 

CHANGE  OF  EMPHASIS  FROM  NON-HUMAN  TO  HUMAN 
ENERGIES SCIENCE  AS  ENGINEERING  IN  PREPARA- 
TION  ENGINEERING      AS      SCIENCE      IN      ACTION 

MATHEMATICS  THE  GUIDE  OF  THE  ENGINEER EN- 
GINEERING  THE   GUIDE   OF    HUMANITY HUMANITY 

THE  CIVILIZING  OR  TIME-BINDING  CLASS  OF  LIFE — ' 
THE  FOUR  DEFINING  MARKS  OF  THE  GREAT  ENGI' 
NEER  OF  THE  FUTURE ENGINEERING  STATESMAN- 
SHIP. 

I  AM  not  a  professional  engineer.  What,  then,  is  my 
apology  for  daring  to  speak  of  engineering?  It  is  not, 
I  fear,  a  quite  convincing  one.  For  it  is  the  apology  of 
a  layman  who  can  only  plead  that  for  more  than  twenty- 
five  years  he  has  taught  mathematics  to  engineering  stu- 
dents; that  during  these  years  he  has  associated  a  good 
deal  both  with  practicing  engineers  and  with  professors 
of  engineering  science  and  art;  that,  like  all  who  think 
of  the  matter,  he  has  been  deeply  impressed  in  beholding 
and  contemplating  engineering  achievements,  from  the 
great  pyramids  and  aqueducts  and  roads  of  what  we  call 
antiquity  down  to  the  rapidly  multiplying  marvels  wrought 
on  every  hand  by  the  engineering  prowess  of  our  own  day; 
that  he  has  examined  some  of  the  writings  of  engineers, 
ancient,  mediaeval,  and  modern — the  work  of  Frontinus, 

452 


SCIENCE  AND   ENGINEERING  453 

the  engineering  speculations  of  Leonardo  da  Vinci,  es- 
pecially the  famous  books  recently  produced  by  "wizard" 
apostles  of  "efficiency";  and  that  he  has  been  thus  led 
to  reflect  a  good  deal  upon  the  opportunities,  the  func- 
tions and  the  obligations  of  engineering,  rightly  conceived, 
in  the  great  affairs  of  our  human  world.  There  is,  more- 
over, the  general  consideration  that  a  layman,  viewing 
a  profession  from  the  outside,  seeking  thus  to  ascertain 
its  proper  relations  to  the  common  weal,  may  bring  to  the 
task  a  certain  freedom,  which,  were  he  a  member  of  the 
profession,  he  might  have  lost.  "Men  trained  in  a  pro- 
fession," said  Professor  David  Swing,  "come  by  degrees 
into  the  profession's  channel,  and  flow  only  in  one  direc- 
tion, and  always  between  the  same  banks.  The  master 
of  a  learned  profession  at  last  becomes  its  slave.  He  who 
follows  faithfully  any  calling  wears  at  last  a  soul  of 
that  calling's  shape.  You  remember  the  death  scene  of 
the  poor  old  schoolmaster.  He  had  assembled  the  boys 
and  girls  in  the  winter  mornings  and  had  dismissed  them 
winter  evenings  after  sundown,  and  h^d  done  this  for 
fifty  long  years.  One  winter  morning  he  did  not  appear. 
Death  had  struck  his  old  and  feeble  pulse;  but,  dying, 
his  mind  followed  its  beautiful  but  narrow  river-bed,  and 
his  last  words  were:  'It  is  growing  dark — the  school  is 
dismissed — let  the  girls  pass  out  first.'  "  Finally,  it  is  not 
my  intention  to  deal  with  the  technique  of  engineering  nor 
with  that  of  any  branch  thereof,  but  rather  with  its  gen- 
eral aspects,  with  what  is  essentially  common  to  its 
branches,  with  the  science  viewed  as  a  whole.  I  shall 
not  be  so  much  concerned  with  the  present  status  of  the 
science  as  with  its  potence  and  promise.  Of  individual 
engineers  the  ideals  may  be  high  or  low,  worthy  or  un- 


454  MATHEMATICAL   PHILOSOPHY 

worthy;  but  of  engineering  itself  the  ideal  is  great  and 
mighty.     It  is  of  that  ideal  that  I  intend  to  speak. 

What  is  engineering?  It  is  evident  that  the  term 
stands,  or  ought  to  stand,  for  a  great  conception.  What 
is  that  conception?  Many  attempts  have  been  made  to 
define  it.  Most  of  them  throw  more  light  upon  the  char- 
acter and  outlook  of  those  who  have  made  them  than 
upon  the  nature  of  engineering  itself. 

To  say  that  an  engineer  is  one  who  "knows  what  to 
do,  when  to  do  it  and  how  to  do  it"  may  be  true, — the 
formula  is  very  neat, — but  it  can  hardly  be  said  to  be 
quite  definitive,  seeing  that  it  applies  equally  well  to  the 
wisdom  of  a  wise  philanthropist  and  to  the  cunning  of 
a  cunning  thief. 

To  define  engineering  in  terms  of  aim  is  no  doubt 
feasible;  but  to  say  that  the  aim  is  "maximum  produc- 
tion with  minimum  outlay  of  time,  effort  and  resources" 
sounds  like  the  "efficiency"  cry  of  brute  industrialism, 
appears  to  regard  quantity  as  the  summum  bonum,  seems 
to  ignore  the  spiritual  autonomy  of  men  and  women,  and 
to  idealize  a  "system"  in  which  "laborers"  are  reduced 
to  the  level  of  machines. 

To  say  that  the  aim  of  engineering  is  the  "mastering 
of  natural  forces  and  materials  for  the  benefit  of  man- 
kind" is  far  better  in  one  respect  because  it  is  humane — 
it  represents  engineering,  I  mean,  as  having  for  its  aim 
"the  benefit  of  mankind."  But  what  do  its  sponsors  mean 
by  "natural  forces"?  Do  they  intend  the  term  to  cover 
the  personalities  of  individual  men  and  women,  their 
perfectly  natural  civilizing  impulses  and  aspirations?  Do 
they  include  among  "natural  forces"  the  spiritual  energies 
of  our  human  kind — those  time-binding  powers  in  virtue 
of  which  human  beings  are  human?    If  they  do  not,  why 


SCIENCE  AND  ENGINEERING  455 

not?  And  if  they  do,  what  do  they  mean  by  "the  master- 
ing of  natural  forces"  ?  The  questions  are  important  and 
sometime  the  philosophers  of  engineering  must  answer 
them. 

The  most  famous  conception  of  engineering  and,  in 
the  judgment  of  many,  the  best  one  to  be  found  in  the 
literature  is  almost  a  century  old.  It  is  due,  I  believe, 
to  the  English  engineer,  Thomas  Tredgold  (i  788-1 829) 
and  is  found  in  the  charter  of  the  Institution  of  Civil 
Engineers  (London,  1828).  Engineering  is  there  called 
an  art — "the  art  of  directing  the  great  sources  of  power 
in  nature  for  the  use  and  convenience  of  man,  as  the 
means  of  production  and  of  traffic  in  states,  both  for 
external  and  internal  trade,  as  applied  in  the  construction 
of  roads,  bridges,  harbours,  moles,  breakwaters,  and 
lighthouses,  and  in  the  art  of  navigation  by  artificial 
power  for  the  purposes  of  commerce,  and  in  the  drainage 
of  cities  and  towns."  The  gist  of  the  matter  is  in  the  first 
eighteen  words:  the  art  of  directing  the  great  sources  of 
power  in  nature  for  the  use  and  convenience  of  man.  For 
our  purpose  it  will  be  well  worth  while  to  reflect  upon 
them  a  little.  Though  found  in  a  charter  for  civil  as 
distinguished  from  military  engineering,  they  apply  no 
better  to  what  we  today  call  civil  engineering  than  to  any 
other  of  the  numerous  varieties  into  which,  since  the 
words  were  written,  engineering  has  branched;  moreover, 
they  apply  no  better  to  a  branch  of  engineering  than  to 
Engineering  itself,  regarded  as  one  great  Enterprise,  and 
that  is  why  it  will  repay  us  to  reflect  upon  them. 

In  view  of  their  date  (1828)  it  is  not  strange  that 
they  represent  engineering  as  an  "art"  instead  of  a  science, 
as  we  call  it  today,  or  a  science  and  art,  as,  I  think,  we 
ought  to  call  it.    But  that  is  a  trivial  matter. 


456  MATHEMATICAL   PHILOSOPHY 

What  is  not  trivial, — what  is  indeed  of  the  gravest 
importance, — are  the  major  emphases  in  the  conception. 
These  are  two:  one  of  them  is  upon  the  ultimate  aim, 
purpose,  or  end  of  engineering;  the  other  is  upon  means 
thereto. 

What  is  the  former?  .What  is  engineering  for?  Is 
it  for  "the  use  and  convenience"  of  engineers?  Or  for 
that  of  a  "shop"?  Or  that  of  a  manufacturing  estab- 
lishment? Or  that  of  an  industry?  Or  that  of  a  group 
of  "capitalists"?  Or  that  of  a  class  of  "laborers"?  No; 
it  is  for  no  such  restricted  good;  it  is  infinitely  higher 
and  nobler  and  more  embracing — engineering  is  for  "the 
use  and  convenience  of  man''';  and  "man"  does  not  mean 
this  group  or  that;  it  means  all  the  people  of  the  world, 
not  only  those  now  living  but  an  unending  succession  of 
generations  to  come.  The  appeal  is  thus  to  an  imagina- 
tion great  enough  to  grasp  and  represent  the  race.  We 
are  wont  to  say  that  in  things  human  there  can  be  no 
perfection.  I  believe  we  may  say,  however,  that  the  ulti- 
mate aim,  purpose,  or  end  of  engineering  as  presented  in 
that  century-old  conception  of  it  is  a  perfect  ideal  and 
can  never  be  improved. 

But  of  the  other  major  emphasis  in  the  famous  state- 
ment the  same  cannot  be  said — far  from  it.  For  note  its 
incidence.  Where  does  the  emphasis  fall?  It  falls  upon 
"directing  the  sources  of  power  in  nature,"  and  the  an- 
swer is  important  as  indicating  the  psychology,  the  at- 
titude and  temper,  the  social  philosophy,  of  an  age — an 
age  that  still  lingers,  but  is  being  outgrown  and  is  destined 
to  pass.  For  what  is  there  meant  by  "nature"?  It  is 
evident  that  what  is  meant  is  physical  nature,  the  external 
universe,  the  wow-human  part  of  the  world,  and  it  is  evi- 
dent that  the  term  "power"  refers  to  the  blind  forces  of 


SCIENCE  AND  ENGINEERING  457 

that  "nature"— to  wind  and  wave  and  tide  and  gravity 
and  heat  and  so  on.  Such  things,  we  say,  are  non-human, 
— man  has  great  interest  in  them  but  they  have  no  interest 
in  man, — and  when  they  are  made,  as  they  can  be  made, 
to  serve  human  welfare,  what  is  it  that  makes  them  serve? 
Everyone  knows  the  answer:  what  makes  them  serve  is 
human  thought, — it  is  human  intelligence  and  purpose 
and  will, — it  is  the  power  that  invents, — the  power  that 
observes  and  remembers  and  imagines  and  conceives  and 
reasons  and  creates, — it  is,  in  a  word,  what  we  may  for 
convenience  call  the  spiritual  energies  of  our  human  kind. 
These  energies  are  just  as  natural  as  Tredgold's  "power." 
The  reflection  is  no  doubt  just  but  it  is  very  obvious. 
Why,  then,  insist  upon  it  so?  Because,  as  you  must  see, 
it  fundamentally  alters  the  traditional  point  of  view.  We 
are  seeking  a  just  and  worthy  conception  of  the  science 
and  art  of  Engineering,  and  the  reflection  in  question 
radically  shifts  the  incidence  of  the  major  emphasis.  It 
shifts  it  from  the  non-human  to  the  human.  For  it  is 
clear  that  what  requires  "directing" — what  requires  to 
be  engineered — is  primarily,  not  the  blind  forces  of  ex- 
ternal nature,  but  those  other  natural  forces — the  spirit- 
ual energies  of  Man.  It  is  perfectly  evident  that  the 
ultimate  aim  and  ideal  of  engineering, — the  welfare  of 
our  human  kind, — not  only  demands  the  conquest  of 
physical  nature,  not  only  demands  subjugation  of  the  non- 
human  forces  of  the  world,  but  also  demands,  as  even 
more  essential,  world-wide  enlightenment  of  human 
beings,  world-wide  coordination  of  human  effort,  world- 
wide establishment  of  Justice;  and  it  is  perfectly  evident 
that  the  sole  means  to  these  great  ends  is  the  understand- 
ing and  "directing," — the  "engineering,"  if  you  please, 
— of  what  we  have  called  the  spiritual  energies  of  man. 


458  MATHEMATICAL   PHILOSOPHY 

These  are  the  energies  with  which  we  dealt  in  the 
preceding  lecture;  they  are  the  energies  which,  in  Korzyb- 
ski's  fine  phrase,  constitute  humanity  the  "time-binding" 
class  of  life;  they  are  the  human  energies  in  virtue  of 
which  the  distinctive  life  of  man  is  life-in-time;  they  are 
the  energies  that  make  man  the  creator  of  Civilization; 
man  is  their  sole  agency,  their  sole  instrument,  their  sole 
organ;  characteristic  of  humankind,  they  are  present  in 
some  measure  wherever  human  beings  are  found.  Upon 
the  effectiveness  of  these  energies  depends  the  creation 
of  material  and  spiritual  wealth — the  advancement  of 
civilization — the  well-being  of  man.  To  be  effective, 
however,  they  must  be  understood,  they  must  be  organ- 
ized, they  must  be  coordinated,  they  must  be  brought  into 
world-wide  cooperation — in  one  word,  they  require  to  be 
engineered.  And  so  I  propose  to  define  Engineering  to 
be 

The  science  and  art  of  directing  the  time-binding 
energies  of  mankind, — the  civilizing  energies  of  the 
world, — to  the  advancement  of  the  welfare  of  man.1 

That  conception  does  not  represent  engineering  as  jt 
has  been  practiced  in  the  past  nor  as  it  is  practiced  today. 
It  represents  an  Ideal  which  engineering  will  approxi- 
mate more  and  more  just  in  proportion  as  it  becomes  more 
and  more  humanized  and  enlightened.  The  ideal  is  an 
inspiring  one;  but  it  ought  not  to  flatter  the  vanity  of 

1  My  friend,  Mr.  Robert  B.  Wolf,  has  pointed  out  to  me  that  the 
preamble  of  the  Constitution  (1920)  of  The  Federated  American  Engi- 
neering Societies  says:  "Engineering  is  the  science  of  controlling  the 
forces  and  of  utilizing  the  materials  of  nature  for  the  benefit  of  man, 
and  the  art  of  organizing  and  of  directing  human  activities  in  connection 
therewith."  I  hope  the  reader  will  compare  that  conception  critically 
with  the  one  which  I  have  submitted.  The  preamble  dedicates  the 
federation  "to  the  service  of  the  community,  state,  and  nation."  Why  not 
to  the  service  of  the  World? 


SCIENCE  AND  ENGINEERING  459 

professional  engineers;  it  ought  rather  to  give  them  a 
feeling  of  humility.     For  consider  its  spirit  and  its  scope. 

Its  spirit  is  not  a  self-serving  spirit  nor  a  class-serving 
spirit  nor  any  provincial  spirit;  it  is  a  world-serving 
spirit — the  spirit  of  devotion  to  the  well-being  of  all 
mankind  including  posterity. 

And  what  is  its  scope?  Is  it  confined  to  the  kinds 
of  work  done  today  by  professional  engineers  in  the  name 
of  engineering?  It  is  by  no  means  thus  confined;  its 
scope  is  immeasurably  greater;  for,  over  and  above  such 
work,  which  no  one  could  wish  to  belittle,  it  embraces 
whatever  may  be  intelligent,  humane,  and  magnanimous 
in  the  promotion  of  science,  in  the  work  of  educational 
leadership,  in  the  conduct  of  industrial  life,  in  the  estab- 
lishment and  administration  of  justice — in  all  the  affaire 
of  a  statesmanship  big  enough  to  embrace  the  world. 

I  am  facing  the  future,  and  I  say  "in  all  the  affair* 
of  statesmanship"  because  I  do  not  doubt  that  the  affairs 
of  state, — which  are  the  affairs  of  man, — will  at  length 
be  rescued  from  the  hands  of  "politicians"  and  be  com- 
mitted to  a  statesmanship  which  will  be  an  engineering 
statesmanship  because  it  will  guide  itself  and  the  affairs 
of  state  in  scientific  light  by  scientific  means. 

Engineering  statesmanship  will  know  enough  to  know 
that  scientific  knowledge  cannot  be  applied  to  the  conduct 
of  human  affairs  if  such  knowledge  does  not  exist;  it  will 
have  sense  enough  to  know  also  that  knowledge  which 
does  not  exist  cannot  be  suddenly  called  into  existence  at 
the  moments  when  it  is  needed.  Engineering  statesman- 
ship will,  therefore,  be  sagacious  enough  to  make  ample 
provision  in  advance  for  scientific  research;  not  only  for 
technological  research,  but, — primarily  and  especially, — 
for  that  kind  of  research  which  does  not  consciously  aim 


460  MATHEMATICAL   PHILOSOPHY 

at  utility  or  applications.  What  kind  is  that?  It  is  the 
kind  whose  "only  purpose,"  in  the  clear  words  of  Presi- 
dent Nichols,1  "is  the  discovery  of  new  knowledge  with- 
out thought  of  any  material  benefit  to  anybody" ;  it  is  the 
kind  which  Simon  Newcomb  2  had  in  mind  when  he  said 
that,  "The  true  man  of  science  has  no  such  expression  in 
his  vocabulary  as  'useful  knowledge'  ";  it  is  the  kind  of 
which  Henri  Poincare  said  that,  if  there  can  be  "no 
science  for  science's  sake,"  there  can  be  "no  science";3 
it  is,  in  a  word,  the  kind  of  research  which  springs  out 
of  pure  scientific  curiosity, — out  of  wonder,  as  Aristotle 
said, — and  which,  just  because  it  is  thus  disinterested, 
just  because  it  seeks  the  True,  is  the  principal  source  of 
the  Useful  also. 

The  subject  of  such  research  will  be  Nature, — non- 
human  nature  and  human  nature, — the  nature  of  the  non- 
human  world  and  the  nature  of  man, — for  we  can  know 
nothing  else.  Engineering  statesmanship  will  have  sense 
enough  to  know  that  its  work  cannot  be  done  without 
scientific  knowledge  of  both  kinds  of  nature;  it  will, 
therefore,  provide  every  means  for  promoting  the  ad- 
vancement of  the  physical  sciences  and  of  those  biological 
sciences  that  deal  with  the  non-human  world;  and  it  will 
especially  provide  every  means  for  promoting  those  re- 
searches which  have  for  their  aim  the  understanding  of 
Man.  I  have  said  "especially"  because  engineering  states- 
manship will  have  sense  enough  to  know  that,  of  all  the 
things  it  must  deal  with,  man  is  the  supreme  reality,  and 
that,    therefore,    the    understanding    of    man, — scientific 

1  The  Inaugural  Address  of  the  President  of  the  Massachusetts  Insti- 
tute of  Technology,  Science,  June  io,  1921. 

2  Congress  of  Arts  and  Sciences,  vol.  I,  p.  137. 

*  Poincare:   Science   and  Method    (translation   by   Francis    Maitland), 
p.  1 6, 


SCIENCE   AND   ENGINEERING  461 

knowledge  of  human  nature, — is  absolutely  essential  to 
its  enterprise. 

And  here  we  must  say  a  word  respecting  the  relation 
of  engineering,  as  it  is  here  conceived,  to  Education.  The 
science  and  art  of  human  engineering, — the  science  and 
art  of  engineering  statesmanship, — is  based  upon  a  most 
important  assumption.  The  assumption  is  that  when 
and  only  when  men  and  women  are  everywhere  bred  in 
the  knowledge  and  the  feeling  of  what  man  distinctively 
and  naturally  is,  it  will  be  possible  so  to  organize,  to  co- 
ordinate, and  to  direct  the  time-binding  powers  of  man- 
kind,— the  civilizing  energies  of  the  world, — that  Civi- 
lization will  advance  in  accord  with  its  natural  law,  which 
is  that  of  a  swiftly  increasing  exponential  function  of 
Time.  And  so  engineering  statesmanship  will  not  only 
provide,  as  said,  for  the  scientific  study  of  man,  but  it  will 
provide  a  system  of  education  whereby  the  children  of 
man  will  be  taught  the  results  of  such  study, — an  educa- 
tion which  will  have  for  its  supreme  obligation  to  teach 
boys  and  girls  and  men  and  women  what,  as  representa- 
tives of  man,  they  really  and  naturally  are, — not  a  higher 
species  of  animal,  nor  a  lower  species  of  angel, — but 
humans,  whose  proper  life  is  time-binding  life,  civilizing 
life,  life-in-Time. 

In  view  of  such  considerations  it  is  easy  to  see  what 
the  defining  marks  of  a  great  engineer  are  destined  to 
be.  They  will  not  be  the  marks  of  mere  "efficiency"  nor 
of  mere  technological  knowledge  nor  of  technological 
skill — they  will  not  be  mere  engineering  technique  of  any 
kind,  whether  "civil"  or  "mechanical"  or  "marine"  or 
"architectural"  or  "sanitary"  or  "chemical"  or  ''elec- 
trical" or  "industrial";  these  things  will  be  important,  as 
they  are  now,  they  will  indeed  be  indispensable,  but  they 


462  MATHEMATICAL   PHILOSOPHY 

will  not  constitute,  and  they  will  not  define,  the  great 
engineer.  The  characteristic  marks  of  the  great  engineer 
will  be  four:  Magnanimity — Scientific  Intelligence — Hu- 
manity— Action. 

He  will  be  religious  and  he  will  be  patriotic:  "to  do 
good"  will  be  his  religion,  and  his  love  of  country  will 
embrace  the  world.  For  he  will  be  the  scientific  organizer 
and  director  of  the  civilizing  energies  or  the  World  in 
the  interest  of  all  mankind.1 

1  Readers  interested  in  what  may  be  called  the  Humanization  of  en- 
gineering will  find  it  profitable  to  examine  the  folowing  works: 

F.  W.  Taylor:  The  Principles  of  Scientific  Management  (Harper  and 
Brothers,  1916). 

H.  L.  Gantt:  Work,  Wages,  and  Profits  (The  Engineering  Magazine 
Company,  1916). 

W.  N.  Polakov  and  others:  The  Life  and  Work  of  Henry  L.  Gantt 
(The  American  Society  of  Mechanical  Engineers,   1920). 

Dr.  Walter  N.  Polakov:  Mastering  Power  Production  (The  Engineer- 
ing Magazine  Company). 

Robert  B.  Wolf:  "Individuality  in  Industry"  (Bulletin  of  the  Society 
to  Promote  the  Science  of  Management,  1915)  ;  Non-Financial  Incentives 
(American  Society  of  Mechanical  Engineers,  1918)  ;  "Securing  the  Initia- 
tive of  the  Workman"  (American  Economic  Review  Supplement,  1919)  ; 
and  Modern  Industry  and  the  Individual   (A.  W.  Shaw  Company,   1920). 

Especially  Messrs.  Polakov  and  Wolf  deserve  the  highest  commenda- 
tion and  the  thanks  of  all  men  and  women  for  their  insistence  upon 
bringing  the  theory  and  practice  of  engineering  under  the  control  of 
humane  considerations,  upon  basing  engineering  on  the  time-binding 
principles  characteristic  of  humans  instead  of  the  space-binding  principles 
characteristic  of  animals,  and  upon  thus  making  engineering  the  chief 
of  civilizing  agencies,  devoted  to  the  promotion  of  Freedom  and  Justice 
throughout  the  World.  These  men  have  the  vision  to  see  that  the  time 
is  coming  when  to  call  one  a  "mere  space-binder"  will  be  to  call  him 
a  brute  but  to  call  one  a  time-binder  will  be  to  call  him  a  man,  a  human. 


INDEX 


Abel,  207,  209,  212 
Archimedes,  81,  101,  305 
Aristotle,  27,  29,  36,  42,  227,  230, 

321,  391,  397,  405,  460 
Arnold,  9 
Aronhold,  194 

Bacon,  132 
Bailey,  231,  306 
Balzano,  158,  159,  303 
Beman,  393 
Bentley,  315 
Bergson,  33,  34 
Blackstone,  134 
Bliss,  123 
Bolton,  140 
Bolyai,  351 
Boole,  29,  193 
Broad,  140 
Bryan,  151 

Cantor,  298,  303,  308,  393,  413 

Carlyle,  442 

Cauchy,  332 

Cayley,  193,  194 

Chasles,  333 

Cicero,  134,  232 

Clebsch,  194 

Clifford,  140,  232,  333,  349,  354, 

400 
Coke,  134 


Cole,  340 

Compte,  135,  349 

Cousin,  33 

Couturat,  29,  124,  126,  349 

Croce,  35,  321 

Darboux,  400 

Darwin,  151 

Daubeny,  315 

Da  Vince,  227,  453 

Day,  141 

Dedekind,  303,  393 

De  Morgan,  29,  164 

Desargues,  63,  342 

Descartes,  8,  36,  63,  73,  150,  321 

Dewey,  128,  361 

Dickson,  123 

Du  Bois-Reymond,  400,  401 

Eddington,  180 
Einstein,  151 
Eisenstein,  194,  363 
Empedocles,  229 
Enriques,  41,  42,  126 
Epicurus,  231,  409 
Euclid,  43,  44,  54,61,80,  102,  112, 
143.  35I.364 

Fechner,  367,  368,  370,  372,  .375- 

376,  377.  38o,  376 
Fields,  153 


463 


464 


INDEX 


Flammarion,  349 
Foster,  232 
Frege,  29,  178 
Frontinus,  453 

Galileo,  8,  304,  349 
Gantt,  462 
Gauss,  134,  193 
Gergonne,  342,  343,  350 
Gladstone,  51 
Goethe,  24 
Gordan,  194 

Haeckel,  349 

Haldane,  232,  439 

Halsted,  352 

Hamilton,  35 

Hardingham,  232 

Hardy,  400 

Heath,  41,  351,  355 

Hedrick,  125 

Helmholz,  140 

Heracleitus,  229 

Herbart,  368,  369 

Hermite,  194 

Herschel,  9 

Hilbert,  29,  43,  44,  47,  54,  58,  60, 

61,  66,  71,  80,  96,  102,  115,  121, 

124,  163 
Holgate,  123 
Hume,  321 

Huntington,  29,  123,  126,  128,  298 
Huxley,  9,  140,  349 

Inge,  232 
Ingold,  125 

James,  35,  137,  179,  361,  367,  368, 
375>  376,  378,  379 


Jeans,  338 
Jevons,  29 
Jowett,  437 
Justinian,  134 

Kanadi,  315 

Kant,  321,  334,  354,  368,  369 

Kelvin,  9,  344 

Keyser,  179,  369,  370 

Klein,  355 

Kline,  126 

Koo,  87 

Korzybski,  10,  185,  422,  425,  427, 

430,  431,  433,  435,  438,  440,  442, 

445 

La  Farge,  197 

Lagrange,  134,  193,  227 

Lambert,  29,  164 

Le  Conti,  348 

Leibniz,  8,  29,  36,  50,  227 

Lenin,  145 

Lobachevski,  80,  123,  351,  352 

Lodge,  13, 

Lorentz,  176 

Lucretius,  231,  300,  304,  306,  308, 

310,312,314,409 
Lyell,  232 

MacCall,  29 
Mach,  140,  149,  349 
MacMahon,  196,  197 
Mahomet,  145 
Maitland,  462 
Manning,  405 
Martineau,  135 
Marx,  151 
Masson,  306 
Maxwell,  314 


INDEX 


4G5 


Merz,  137 
Miller,  123 
Milton,  3 

Minkowski, '176,  177 
Mobius,  407 
Moleschott,  232 
Montesquieu,  134 
Moore,  126 
Moritz,  24 
Munro,  231,  306 

Newcomb,  460 

Newman,  9 

Newton,  9,  36,  149,  175,  177,  315 

Nichols,  460 

Novalis,  24 

Pardoa,  126 

Paris,  145 

Pascal,  29,  42,  63,  321 

Pasch,  65,  79 

Peano,  29,  126,  1 27,  17S 

Peirce,  C.  S.,  29,  164,  361 

Peirce,  Benj.,  134 

Perry,  321 

Pieri,  122,  124,  126,  134 

Plato,  8,  19,  21,  22,  24,  27,  29,  36, 

4i>  *35>  I5i>358 
Plucker,  113,  329 
Poincare,  21,  42,    134,   140,   170, 

223,  301,  358, 3g6, 398>  4°4>  4H. 

436 
Polakov,  462 
Ptolemy,  405 
Pythagoras,  8,  390 

Rankine,  196,  197 

Reynolds,  197 

Riemann,  80,  123,  352,  368,  378, 


Ritter,  425 

Ross,  390 

Royce,  298,  349 

Riickert,  232 

Rudolph,  406 

Ruffini,  227 

Russell,  25,  29,  30,  49,  126,  127, 

133,    136,    164,    178,    179,   225, 

240,  370,  390,  395 

Salmon,  333 

Santayana,  320 

Schiller,  55 

Schoute,  337 

Schopenhauer,  35 

Schroder,  29 

Scripps,  348 

Shaw,  348 

ShefFer,  126 

Simplicius,  405 

Slosson,  349 

Smith,  D.  E.,  30,  123,  393 

Smith,  W.   B.,  36,  134,  140,  233, 

335,400,  411 
Socrates,  42 
Sommerville,  318 
Spencer,  228,  230 
Spinoza,  36,  321 
Stout,  379 
Stifel,  406 
Swing,  453 
Sylvester,  24,  194,  298,  333,  393 

Taylor,  462 
Thales,  8 
Titchener,  379 
Townsend,  44 
Tredgold,455,457 
Tyndall,  349 


466 


INDEX 


Vaihinger,  334 

Vailati,  126 

Veblen,  62,  122,  123,  124,  128,  336 

Virgil,  232 

Wagner,  152 

Wallis,  406 

Ward,  379 

Weber,  83,  358 

Weber,  E.  H.,  367,  368,  370,  372, 

396 
Weierstrass,  393 
Wellstein,  83,  358 


Whitehead,  14,  29,  30,  125,  136, 

164,  178,  179,  240 
Williams,  448 
Wolf,  458,  462 
Woods,  123,  355 
Wundt,  375 

Young,  J.  W.  A.,  122,  123 
Young,  J.  W.,  62,  124,  129,  298, 
336 

Zeller,  229 
Zeuxis,  197 


tjX'ij 


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